In a large skillet over medium heat, melt the butter then add the vegetables and bacon. Stirring often, saute the bacon and vegetables in the butter for about 20 minutes, until the bacon starts to crisp on the edges and the vegetables begin to caramelize.
Shift the mixture in the pan to evenly distribute it in a layer across the surface, then make a small well in each quarter section. Break one egg into each well. Continue cooking until the eggs are almost done. If you don’t mind cooked yolks, just put a lid over the skillet and let the eggs steam until cooked through. (I like my yolks runny, so I let the whites cook enough until I can turn the eggs over and quickly finish the whites, leaving the yolk still liquidy.)
When the eggs are almost done, sprinkle the cheese over the top and let cook a little more until the cheese melts. Serve hot.
