Mash bread and milk in large bowl with fork until homogeneous (you should have about 1/2 cup). Stir in salt, pepper, garlic, and steak sauce.
Break up meat into small pieces over bread mixture. Using fork or hands, lightly mix together until mixture forms cohesive mass. Divide meat into 4 equal portions. Gently toss one portion of meat back and forth between hands to form loose ball. Gently flatten into 3/4" thick patty that measures about 4 1/2" in diameter. Press center of patty down with fingertips until it is about 1/2" thick, creating slight depression in each patty. Repeat with remaining portions of meat.
LIghtly dip wad of paper towels in vegetable oil; holding wad with toongs, wipe grill cooking grate. Grill burgers on hot side of grill, uncovered, until well seared on first side, 2-4 minutes. Using wide metal spatula, flip burgers and continue grilling, about 4 minutes. Add cheese about 2 minutes before reaching desired doneness, covering burgers with disposable aluminum foil to melt cheese.
While burgers grill, toast buns on cooler side of grill. Serve on buns with desired toppings.