Preheat broiler. Cook garlic in oil in a 10-inch well-seasoned cast-iron or other ovenproof skillet over moderate heat, stirring occasionally, until golden, about 2 minutes. Discard garlic and add arugula, then cook, stirring frequently, until wilted, 1 to 2 minutes. Whisk together eggs, salt, and pepper until combined, then pour over arugula in skillet and cook, undisturbed, over moderate heat until almost set, 5 to 6 minutes. Sprinkle cheese evenly on top and broil 4 to 5 inches from heat until eggs are just set and cheese is melted, 1 to 2 minutes.