Smoky Olive-Oil Fried Eggs

Preheat broiler. Mince and mash garlic to a paste with a pinch of salt, then stir together with red-pepper flakes and 2 tablespoons oil in a small bowl. Break eggs into a bowl. Heat remaining 2 tablespoons oil in a flameproof 12-inch heavy skillet over medium-high heat until it shimmers. Pour in eggs and cook, undisturbed, until whites begin to set, then cook, lifting up edge of cooked whites with a spatula to let as much raw egg white as possible flow underneath, 2 minutes (top and yolks will still be very loose). Remove skillet from heat and sprinkle cheese over eggs. Broil 4 to 6 inches from heat until cheese is melted and bubbling and yolks are barely set, 2 to 4 minutes. Serve drizzled with garlic oil.