@Buffy is right that you're unlikely to be able to influence processing speed for a given journal.
One of the axes on which journals compete is their processing speed. SciRev provides a platform for comparison (e.g. here are the reviews for statistics journals, sorted in increasing order of "total handling time":
note that reviews are provided by authors, and the number of reviews per journal varies enormously, so you should be appropriately careful interpreting the results). Googling "academic journal processing time" points you to a variety of other discussions/comparisons. Sometimes journals provide this information on their web pages.
Thus you could aim for a venue with rapid turnaround. Unfortunately for you, in my experience there's a correlation between selectivity/flashiness and rapid turnaround (e.g. high-impact journals often ask for reviewers to return reviews on a very short time scale), so you might have trouble getting your paper accepted in such a publication. It may be that some of the less-traditional, more open venues such as PeerJ have relatively fast turnaround times.
You obviously know the policies of your advisor/institution better than we do, but (again in my experience, in not totally unrelated fields) it's often sufficient to have a paper submitted to a reputable journal for it to count toward a dissertation; this rule prevents the committee from having to deal with crappy manuscripts in an early stage of preparation, but insulates the students from the vagaries of the publication process. I'd definitely recommend double-checking with someone knowledgeable ...