Follow these simple steps to prepare your German Potato Omelet:

1. Prepare the Potatoes
   In a large skillet, cook the thinly sliced potatoes in 2 tablespoons of butter over medium heat for about 15 minutes. Stir occasionally until the potatoes are browned and tender. Add the sliced green onions and mix well. Keep the mixture warm.
2. Prepare the Egg Mixture
   In another nonstick skillet, melt the remaining butter over medium-high heat. In a bowl, whisk together the eggs and milk until well combined.
3. Cook the Omelet
   Pour the egg mixture into the preheated skillet. Let it cook until the edges start to set. Gently push the cooked edges toward the center, allowing the uncooked eggs to flow underneath.
4. Add the Filling
   Once the eggs are nearly set, spread the potato and green onion mixture on one side of the omelet.
5. Fold and Serve
   Fold the other side of the omelet over the filling. Carefully slide the omelet onto a plate, cut it into wedges, and season with salt and pepper to taste.