Wash and dice the Yukon Gold or red potatoes into small cubes. Boil in salted water until fork-tender, about 10-15 minutes. Drain and set aside.

Melt butter in a large skillet over medium heat. Add the drained potatoes and cook until golden brown and crispy, about 5-7 minutes.

If using, chop ham or bacon and add to the skillet with the potatoes. Cook for an additional 2-3 minutes until heated.

In a medium bowl, whisk together eggs, milk, salt, and pepper until smooth. This creates a fluffy texture for the scramble.

Pour the egg mixture over the potato and meat in the skillet. Cook for 1-2 minutes undisturbed, then gently stir. Cook on medium-low for another 3-5 minutes until eggs are set but creamy.

Sprinkle shredded cheddar cheese over the eggs, cover, and cook for another 2-3 minutes until cheese is melted.

Remove from heat and garnish with green onions or chives before serving warm.