In a medium bowl, whisk together the eggs, milk, salt, and pepper until smooth and well combined.
In a large skillet, melt the butter over medium heat. Add the diced potatoes and cook until they are tender and golden brown, about 10-15 minutes.
If using, stir in the cooked chicken or beef and continue to cook until heated through.
Pour the egg mixture over the potatoes in the skillet. Cook, stirring gently, until the eggs are scrambled and cooked to your liking.
Remove from heat and stir in the shredded cheddar cheese and chopped green onions or chives until the cheese is melted. Serve warm.