Dice the potato and onion. In a medium bowl, whisk together the eggs, milk, salt, and pepper. Set aside.

Heat olive oil in a large non-stick skillet over medium-high heat. Add the diced potatoes in a single layer. Cook for 10-12 minutes, stirring occasionally, until golden brown and crispy on all sides.

Add the diced onion to the skillet with the potatoes. Cook, stirring, for 2-3 minutes until the onion is softened and fragrant.

Reduce heat to medium-low and pour the whisked egg mixture over the potatoes and onions. Let the eggs set for about 30 seconds before gently pushing them across the pan with a spatula to form soft curds.

When the eggs are about 80% cooked, sprinkle the shredded cheddar cheese over the top. Gently fold everything together until the cheese is melted and the eggs are cooked through. Remove from heat, garnish with fresh chives, and serve immediately.