Preheat oven to 350 degrees F (175 degrees C). Lightly grease an 8-inch square baking dish.
Beat eggs together in a large bowl. Stir milk into eggs.
Heat oil in a skillet over medium heat; saute onion until softened, 5 to 10 minutes. Add ham; cook and stir until ham is warmed, 2 to 3 minutes. Transfer onion-ham mixture to the prepared baking dish. Layer provolone cheese over onion-ham mixture. Pour egg mixture over cheese; season with salt and pepper.
Bake in the preheated oven until a knife inserted into the center comes out clean, about 45 to 50 minutes.