Add the remaining 2 tablespoons vegetable oil to the skillet and increase the heat to medium high. When the oil is shimmering, add the ramen mixture and spread into an even layer. Cook undisturbed until the bottom begins to turn golden-brown, 2 1/2 to 3 minutes. Pour in the egg mixture over the noodles and shake the pan to distribute it evenly. Cover, reduce the heat to medium, and cook until the egg mixture on top is bubbly and mostly cooked with a few raw spots on top, about 3 minutes.