Heat 2 tablespoons oil in medium skillet over medium-high heat. Add artichokes to skillet; sprinkle with salt and pepper and sauté until beginning to brown around edges, about 4 minutes. Add garlic and 1/2 cup water, reduce heat to medium-low, cover, and simmer until artichokes are tender when base is pierced with small sharp knife, about 10 minutes. Uncover and boil until any liquid remaining in skillet evaporates. Remove from heat.
Preheat broiler. Whisk eggs and 2 tablespoons cheese in large bowl. Sprinkle with salt and pepper. Heat 2 tablespoons oil in medium nonstick ovenproof skillet over medium-high heat. Add mushrooms; sauté until browned, about 5 minutes. Pat artichokes dry; add to skillet and stir 2 minutes. Add eggs; stir to blend. Reduce heat to medium, cover, and cook until eggs are almost set, about 5 minutes. Sprinkle with remaining 2 tablespoons cheese.
Transfer frittata to broiler. Broil just until set in center, about 1 minute. Using rubber spatula, loosen edges of frittata and slide out onto platter. Serve warm or at room temperature.