Do not do a fellowship. The first Q to ask oneself is "what do I want to treat," followed by "how do I want to treat it."
"Therapy" is a very large umbrella. There are technically hundreds of therapies, but for practical purposes, in terms of technique/theory, there's probably less than 20 with all the others being spinoffs or rebrandings of the core ones.
If you're just looking to enhance basic therapy skills, then a psychodynamic training is fine, which most psychoanalytic institutes offer. I'm a fan of Gestalt therapy, which contemporary schools draw from a lot of psychoanalytic theory, but is a bit more human IMHO. There are training institutes in major cities as well. Most people don't really want to be a psychoanalyst (4-5 days a week on the couch). I did a year of analytic training right after residency and it was a helluva lotta work (I had a FT job and was developing another career too).
As Bartelby mentioned you can pay for supervision. At minimum, and I can't overemphasize this, is to get direct feedback of your technique IN session. That means either video recording your sessions or having some two-way mirror sessions. Your benefits from training will at least double. ISTDP training does this as a standard. Otherwise you're talking about what you remember happened, and I guarantee you're missing a thousand things happening in the room (their non-verbals, your non-verbals, other patterns, opportunities for interventions/interpretations/etc that you didn't notice).
One can also consider a supervision/training group. Many headed for private practice fear the isolation, and having a cohort that meets to discuss cases, do trainings together, etc., can help.
Finally many therapies, in terms of the technique, can be learned in workshop format, followed by optional supervision to make sure you're practicing well. At least start with the workshop. I do 1-2 trainings a year, starting in residency, when I went to the Beck Institute for their 2 weeks of training (which I paid for out of pocket). I've made it a point to keep it up, and continually diversify my training. You learn the technique in bursts, and mastery longitudinally. So fellowships IMO are misguided.
I remember a study (which of course I don't have handy), that therapists believe that they continue to improve in isolation, but that they actually stagnate or get worse the longer they practice without supervision/feedback. I'll try to dig it up.