I'm afraid that grep has no such facility. perl and/or python might. Off the top of my head (w/o spending much thought on it) I came up with the following awk-combo.
awk '/Started GET/{print ""};1' blurble | awk 'BEGIN{RS=ORS="\n\n";FS=OFS="\n"}/FATAL/{print $0}'
Given a block of log concocted from your snippet (x4) and a few extra "other log" lines thrown in and two "FATAL" lines removed I get two segments printed:
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] F, [2014-04-10T19:12:36.346380 #1910:e93004] FATAL -- : Error caught: [NoMethodError] undefined method `invert' for nil:NilClass
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] F, [2014-04-10T19:12:36.346380 #1910:e93004] FATAL -- : Error caught: [NoMethodError] undefined method `invert' for nil:NilClass
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)
Running the two awks against the file (I called it blurble) gives this output:
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] F, [2014-04-10T19:12:36.346380 #1910:e93004] FATAL -- : Error caught: [NoMethodError] undefined method `invert' for nil:NilClass
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)
[----] I, [2014-04-10T19:12:36.294512 #1910:e93004] INFO -- : Started GET "/ems_infra/new" for 4.2.2.2 at 2014-04-10 15:12:36 -0400
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] F, [2014-04-10T19:12:36.346380 #1910:e93004] FATAL -- : Error caught: [NoMethodError] undefined method `invert' for nil:NilClass
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.345863 #1910:e93004] INFO -- : other random logs
[----] I, [2014-04-10T19:12:36.369320 #1910:e93004] INFO -- : Completed 200 OK in 69ms (Views: 22.5ms | ActiveRecord: 0.0ms)