Combine the roast, brown sugar, vinegar and water in a 6-qt. cast-iron or other heavy ovenproof pot.
Bake at 375° for 3 hours.
Remove from oven; cool.
Remove all fat and any bones.
Shred beef; return to pot.
Combine ketchup, mustard, onion and garlic; add to beef and stir to blend.
Reduce oven temperature to 300° and cook, covered, for up to 4 hours.
Stir every half hour, adding water or ketchup as needed to keep well moistened.
Serve on buns with onion and cheese.