Set a large cast-iron skillet over medium-high heat. Add bacon lardons when hot, and fry until nearly crisp.
Remove cooked bacon into ramekin or stoneware bowl, along with 1/4 of bacon grease.
Sauté mushrooms and leeks in the greased skillet, stirring often, until both are tender, about three minutes. Add garlic and arugula, and sauté for one minute, stirring constantly.
Remove cooked ingredients into ramekin with bacon. Form an indent in the middle of ramekin, and crack two eggs into depression. Be sure to keep yolks from touching the edge of bowl, or they will set too quickly.
Put ramekin in preheated oven. Bake for ~6 minutes, until egg whites are nearly set.
Remove ramekin and sprinkle Parmesan, cayenne, and black pepper on eggs. Turn on broiler and return ramekin to oven.
Broil for ~2 minutes, until whites have set but yolks still run. Remove and serve immediately.