In a medium sized mixing bowl, toss Brussels with oil, shallot, garlic, sat, pepper, lemon zest and juice.
Heat additional oil or ghee in a large cast iron skillet. Once oil is hot, add Brussels sprouts to skillet and let cook for 3-4 minutes before stirring.
Now toss Brussels around in skillet and let cook an additional few minutes, until crispy and cooked.
Create 4 holes in the Brussels and crack an egg in each hole. Cook for 4-6 minutes, or until egg whites are set. (Helps to cover the skillet)
Remove from heat, season with salt and freshly cracked pepper and serve with Parmesan and lemon wedges.