Preheat oven to 375°F (190°C). Place a 10-inch cast iron skillet in the oven for 10 minutes.

Carefully remove skillet and add 2 tablespoons butter and olive oil, swirling to coat the pan evenly.

Add shredded potatoes and spread into an even layer. Create three small pockets in the potatoes. Season with salt and pepper.

Dot the surface with 1 1/2 tablespoons butter and return skillet to the oven for 15–18 minutes.

Remove skillet and add 1 teaspoon butter into each potato pocket. Crack one egg into each pocket.

Sprinkle shredded cheese over the potatoes and return skillet to the oven for 8–10 minutes, until egg whites are set but yolks remain slightly runny and potatoes are golden.

Season with salt and pepper, garnish with chives, and serve immediately.