Preheat the oven to 350 degrees F.
In a bowl, beat the egg and 1/2 cup milk. Add the flour and mix well. Add the garlic powder, salt, and pepper. Add the other 1/2 cup milk and mix well. Add the cooked mashed potatoes and mix well. Add the green onion and incorporate into the mixture.
Heat the vegetable oil in a frying pan over medium-high heat. Once the pan is hot, spoon the potato mixture into the pan forming 4 pancakes. Add the butter to the pan. Flatten the pancakes for even cooking, cook 4 minutes, then flip. Place the pan in the oven for 5 minutes, or until the potato pancakes are brown and cooked through.