PREP: Coarsely chop the onion and red pepper; we don't want these finely diced, but chopped to about 1/2-inch pieces. Wash and dry the red potatoes. Peel if desired (I leave the peel on). Cut into small pieces; about 1/2-inch. The potatoes need to be small to get tender in time and the onions and pepper large enough they don't burn in the time it takes to cook the potatoes.
BACON: If using instead of ham, See Note 1.
ONION, PEPPER, AND POTATOES: In a very large nonstick skillet (12 inches) heat 4 tablespoons butter to medium heat. When the butter has melted, add in the prepared red pepper, onion, and potatoes. Season to taste with salt and pepper, and cook over medium heat for about 10-12 minutes, stirring occasionally. If anything begins to burn, turn down the heat; if it's not sizzling and cooking, slightly increase the heat. Once done (crisp on the outside, tender on the inside), empty the veggies onto a plate to sit while we cook the eggs.
EGGS: Combine the eggs, milk (or half-and-half or heavy cream), Cajun seasoning, and salt and pepper to taste. Briskly whisk with a fork to beat air into the eggs and mix until smooth and combined. In the empty skillet, add the remaining 1 tablespoon butter and heat to medium high. Use a silicone spatula to spread the butter to coat the skillet. As soon as the butter is melted, pour the eggs into one even layer. Let stand for 30 seconds without touching. Then use the spatula to push the eggs from one side of the skillet to the other, tipping the skillet so uncooked eggs fill the empty space of the skillet. Continue this process until eggs are cooked, but still slightly wet and shiny. Do not over-cook. As soon as the eggs are set, empty on top of the potatoes. Let pan slightly cool and then return to the burner on low heat.
CHEESE: Transfer all the food from the plate back in the skillet and top with the freshly grated cheese. Stir to melt the cheese. Taste and adjust any seasonings as needed. Transfer to plates and top with chopped bacon or ham.
SERVE: Serve immediately or stuff into tortillas to make breakfast burritos.