Rinse pork and put in a 5- to 6-quart pan. Add onion, tomato, tomato juice, broth, and brown sugar. Rinse dried chili, break off stem, and shake seeds from pod. Add dried chili and chipotle chili to pan.
Cover and bring to a boil over high heat; reduce heat to low and simmer until meat is very tender when pierced, 3 to 3 1/2 hours, occasionally pushing meat aside and stirring sauce to be sure it's not sticking.
Supporting meat with 2 slotted spoons, transfer to an 8- or 9-inch-wide pan, fat up.
Bake meat in a 350° oven until very well browned, 30 to 40 minutes. Cut and discard string. With 2 forks, tear meat into large chunks; keep warm.
Meanwhile, skim and discard fat from meat juices. Boil juices over high heat until reduced to 3 cups, about 30 minutes. Whirl, a portion at a time, in a blender until smooth.
Put pork in a bowl; pour sauce over meat.