Heat oven to 350°F. Brown the sausage in a 12-inch cast iron skillet until fat is rendered and sausage is crumbled and cooked through (if cooked in patties, crumble with your hands once cooled). Transfer to a plate and set aside.
If less than 2 tablespoons rendered fat remain in the skillet, add coconut oil and cook the sweet potato over medium-low heat, stirring frequently, until browned and tender, 10-12 minutes.
(Even if you start with a little fat from the sausage, you may need to add coconut oil if the pan becomes is too dry.) Transfer to a plate and season with salt and pepper.
In the same skillet, saute the onion and bell pepper in the butter over medium heat until softened, about 8 minutes. Add the mushrooms, garlic, and thyme; continue to cook until the mushrooms wilt, about 3 minutes. Stir-in the spinach and cook until the spinach wilts.
Add the crumbled sausage and potatoes back to the skillet, stirring to blend the ingredients. Sprinkle cheese over the top and make 5 small pockets for the eggs. Break 1 egg into each pocket. Season eggs with salt and pepper.
Bake for 10 to 12 minutes or until eggs are just set. Serve immediately with fresh thyme and chopped tomatoes and to garnish.
To save time, use leftover grilled or roasted vegetables, potatoes, or sausages.