Rinse pork and pat dry.
In a blender or food processor, whirl green onions, garlic, chilies, tomato paste, brown sugar, allspice, thyme, 1 teaspoon salt, and 1/2 teaspoon pepper until finely chopped. Add vinegar and whirl until smooth. Scrape mixture into a heavy 5- to 6-quart pan. Add pork and turn to coat completely. Cover and chill at least 2 hours or up to 1 day.
Add 1 cup water to pan, cover, and bring to a simmer over medium-high heat; reduce heat to very low and simmer pork, turning once, until meat is very tender when pierced and shreds easily with a fork, 2 to 2 1/2 hours.
Remove from heat and let cool about 15 minutes. Transfer meat to a bowl. Skim and discard fat from surface of cooking liquid. Measure liquid; if there's more than 1 1/2 cups, boil over high heat until reduced to 1 1/2 cups. With a fork or your fingers, pull meat into thin shreds; remove and discard fat. Mix meat with cooking liquid. Add more salt and pepper to taste.
Spoon about 1/4 cup pulled pork onto each roll bottom; set tops in place.