Preheat the oven to 400°F (200°C). Use a medium oven-safe skillet (or transfer contents to an oven-safe dish before baking).
Place the skillet over medium heat and melt 3 tablespoons butter.
Add 3 cups diced potatoes to the skillet. Cook, stirring occasionally, about 8–10 minutes until the potatoes are tender and start to get slightly crispy.
Add 1 cup chopped red bell peppers and ½ cup chopped onion. Cook, stirring occasionally, 4–6 minutes until the peppers and onion are tender.
Add 1 pound cooked, cubed corned beef and 1 tablespoon Italian seasoning. Stir to combine and cook 3–5 minutes until the corned beef is warmed through and begins to brown slightly.
Make four small wells in the hash and crack the 4 eggs into the wells. Sprinkle ¼ teaspoon salt and ¼ teaspoon pepper evenly over the eggs and hash. Add a pinch of red pepper flakes, if using.
Transfer the skillet to the preheated oven and bake 8–10 minutes, or until the eggs are set to your preferred doneness.
Remove from the oven, let rest 1–2 minutes, then serve.