1. Slice corned beef into 1/2-inch cubes; set aside. Over medium heat, melt 3 tbsp of the butter in a 9- or 10-inch cast-iron skillet. Stir in potatoes, onion, red pepper, garlic, thyme, 1/2 tsp of the salt and the pepper. Mix well and press down with a spatula. Cook 10 minutes. Stir, scraping brown bits from bottom. Press down and cook another 10 minutes. Stir in corned beef, again scraping brown bits from bottom. Press down and cook 10 more minutes.
2. Meanwhile, during the final 10 minutes of cooking time, fry eggs: Heat 1 tbsp of the butter in a large nonstick skillet over medium heat. Crack 3 eggs into skillet, spacing evenly apart. Fry for 3 minutes until whites are set. Repeat with remaining 1 tbsp butter and remaining 3 eggs. Season eggs evenly with remaining 1/4 tsp salt.
3. Serve eggs over hash and, if desired, season with more pepper and chopped parsley.