Sounds like Harry Potter, Heir of Gryffindor by Brent Barten on fanfiction.net, published in 2004. It's pretty long (47 chapters) and focuses on Harry's sixth year.
Harry was ‘professor Potter’ teaching DADA whilst studying.
In chapter 14, "Professor Potter", Dumbledore announces there will be 5 DADA teachers this year; guess who's the fifth?
"Your fifth instructor - for it is not to be me," he
continued, "is a young wizard who has faced dangers most
full-grown wizards could not have handled and come through with
flying colors. And, I am told, both he and his partner are
exceptional duelers. And whether he knows it or not, he is an
excellent teacher who instills confidence in his pupils. He will
primarily be responsible for teaching one of the Tuesday eveming
labs. However, as time allows, he will be circulating through all of
the classes and labs to teach you all a new charm he and two of his
friends have recently developed and assist the other teachers with
their lessons. He needs no further introduction because most of you
already know him." Then, looking squarely into Harry's eyes, he
smiled and said, "Ladies and gentlemen, I give you Professor
Harry Potter."
Harry had a toy horntail dragon that he could control with his mind to practice.
In chapter 30, "First Hogsmeade", Harry has to pick up a toy for mind exercises, and ends up picking up a Hungarian Horntail.
"Yes," the old man said. "He told me that you were
looking for a specific kind of toy, a toy that would challenge your
abilities and force you to think about several things at once, to
make all of the parts work as a whole while keeping each action
separate and unique." [...]
Fifteen minutes later, after carefully examining the few fully
articulating toys in the man's shop Harry selected a half-meter long
model of a Hungarian Horntail - the same dragon he had faced in the
first task of the Tri-wizard Tournament in his fourth year - and
carried it up to the front of the store. The old wizard looked up
when he recognized Harry's selection. "Ah, a little sentimental
value, Mister Potter?" he asked.
The same chapter also mentions a Dr Stone from St Mungo's.
A few minutes later, as the rest of the students were nervously
heading into the Great Hall for dinner, Harry was sitting on a
stretcher on the opposite side of the Entrance Hall with his shirt
off while Dr. Stone was healing a rather nasty cut on his shoulder.
"It's lucky for you my first job at Saint Mungo's was as a
General Healer, Harry," Dr. Stone was saying. "Otherwise I
don't know what I'd do. Do you go looking for trouble or is it
natural?"
Found with the Google query harry potter fanfiction "dr stone" mungo -site:pinterest.*.
