Heat oil in 12-inch non-stick skillet over high heat until smoking. Add brussels sprouts, onions, and potatoes. Stir once then cook without moving until charred, about 2 minutes. Flip, stir, and repeat until well charred all over, about 10 minutes total, adding turkey half way through cooking. Add chili sauce and hot sauce and season to taste with salt and pepper.
Transfer mixture to 10-inch cast iron skillet. Make 2 to 4 wells on surface and add an egg to each. Cover and cook over medium-high heat until bottom is charred and eggs are cooked to desired level of doneness, about 5 to 8 minutes. Serve with turkey gravy.