Preheat oven to 375˚F, for 10 minutes.
Add 2 tablespoons butter and oil to a 10” cast iron skillet and place on the stove over medium-high heat.
Swirl melting butter over the entire surface and add shredded potatoes and garlic and sauté mixture for 3 to 4 minutes. Season with salt and pepper.
Add thyme and chives and continue to sauté for an additional 2 minutes.
Spread mixture into an even layer and create 3 pockets within the potatoes. Season with salt and pepper.
Dot 1 ½ tablespoons butter over the surface and transfer skillet back into the oven for 15-18 minutes.
Remove skillet from the oven and dot each pocket within the potatoes with 1 teaspoon butter. Add a cracked egg into each pocket, top the potatoes with shredded cheese and return the skillet to the oven for 8 to 10 minutes or until the egg whites are set, with the yolks still creamy and the top of the potatoes are golden brown.
Season with salt and pepper and finish with a sprinkle of extra thyme and chives. Serve.