Turn all burners to high, close lid, and heat until very hot, about 15 minutes. Use grill brush to scrape cooking grate clean. Lightly dip wad of paper towels in vegetable oil; holding wad with tongs, wipe cooking grate. Leave primary burner on high, turn other burner(s) to low.
Meanwhile, mash bread and milk in large bowl with fork until homogeneous (you should have about 1/4 cup). Stir in salt, pepper, garlic, and steak sauce.
Break up beef into small pieces over bread mixture. Using fork or hands, lightly mix together until mixture forms cohesive mass. Divide meat into 8 equal portions. Gently toss one portion of meat back and forth between hands to form loose ball. Gently flatten into 1/2-inch-thick patty that measures about 4 1/2 inches in diameter. Press center of patty down with fingertips until it is about 1/3 inch thick, creating a slight depression in each patty. Repeat with remaining portions of meat.
Lightly dip wad of paper towels in vegetable oil; holding wad with tongs, wipe cooking grate. Grill burgers on hot side of grill, covered, until well seared on first side, 2 to 4 minutes. Using wide metal spatula, flip burgers and continue grilling, about 3 minutes for medium-well or 4 minutes for well-done. Serve burgers on toasted buns.