Heat oil in stockpot over medium heat until hot. Place beef roast in stockpot; brown evenly. Remove roast from stockpot.
Add onions to stockpot; cook 8 to 10 minutes or until tender and lightly browned, stirring occasionally. Add water, soy sauce, soup mix and garlic. Return roast to stockpot; bring to a boil. Reduce heat; cover tightly and simmer on range top or in preheated 325°F oven 2½ to 3¼ hours or until roast is fork-tender.
Remove roast; keep cooking liquid warm. Carve roast across the grain into thin slices. Place rolls on metal baking sheet; top evenly with beef. Sprinkle cheese evenly over beef. Place sandwiches under broiler so surface of cheese is 3 to 4 inches from heat. Broil 1 to 3 minutes or until cheese is melted.
For open-faced sandwiches, spoon cooking liquid over top, as desired. For closed sandwiches, serve cooking liquid in small bowls for dipping.