Rinse pork and put in a 6- to 8-quart pan. Add onions, celery, garlic, bay leaves, thyme, 1 teaspoon salt, and enough water to cover meat--2 1/2 to 3 1/2 quarts.
Bring to a boil over high heat; cover, reduce heat, and simmer until meat is very tender when pierced, 2 to 2 1/2 hours. With slotted spoons, transfer pork to a 9- by 13-inch pan; reserve cooking juices. Discard string, and use 2 forks to pull meat into large chunks. Pour milk over meat.
Bake pork in a 325° oven until drippings are browned, about 1 hour, stirring and scraping pan occasionally.
Meanwhile, pour reserved juices through a strainer into a bowl; discard residue. Skim and discard fat. Return juices to pan. Boil over high heat until reduced to 2 cups, about 45 minutes.
When pork drippings are browned, add 1 cup of the reduced juices; scrape drippings free and stir meat, breaking into smaller pieces. Bake until juices have evaporated and drippings are browned, about 15 minutes, stirring occasionally. Repeat step, using remaining juices, and cook until meat edges are crisp and browned, 15 to 20 minutes longer. Season to taste with salt.
Nutritional analysis per 1/2 cup.