Update for anyone who sees this: as of today (2026-11-06):
The process is to go to Migraciones (if you're in CABA, the Sede Central) to the Mesa de Entradas, and then ask for the form to solicit the pronto despacho. The form asks for some basic identifying information (name, DOB, address, expediente number etc) and then there is a section to write your motivos for making the request. You can bring a letter and the clerk will write in this section to see the attached letter. So, essentially what Alby described above.
At least in CABA, there aren't many spots for the mesa de entrada and you can't schedule an appointment for the Sede Central, so you have to show up as early as possible and hope that the check-in guy at the touch screen will give you a number. On a prior visit, this same guy told me they give out about 40 daily spots. I showed up at 6:45a before Migraciones opened around 8a. About 20 people ahead of me in line and by the time the line started moving there were probably 150 people behind me. Finally out by 9:45a.