To make Chili Rub: Stir together aromatic bitters, chili powder and salt; let stand 10 minutes. Cut pork shoulder across grain into 1-1/2- to 2-inch-thick slices; trim excess fat. If needed, cut into chunks to remove some fat.
Rub meat with chili mixture. In 8-quart pot or 10-inch wide high-sided sauté pan with lid, heat oil. Add meat and brown on both sides, about 15 minutes total. Remove meat and place on plate; keep warm.
To make Bitters Sauce: Combine tomato sauce, brown sugar, apple cider vinegar, aromatic bitters, chili powder and garlic cloves; stir well. Pour into hot, empty pan. Simmer 5 minutes, stirring and scraping bottom of pan.
Return meat to pan; spoon sauce over meat. Cover and cook for 75 minutes on low, rearranging meat halfway through cooking. Transfer meat to plate; cover with foil to keep warm.
To thicken sauce, if desired: Let sauce cool in pan 5 minutes then whisk in cornstarch. Bring to boil; continue to cook and whisk until slightly thickened. To serve, pull meat apart with two forks or fingers. Serve warm on buns, drizzled with sauce.