Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the egg noodles, and return to a boil. Cook the pasta uncovered, stirring occasionally, until the pasta has cooked through, but is still firm to the bite, about 5 minutes. Drain well in a colander set in the sink.
Preheat an oven to 350 degrees F (175 degrees C).
Heat 1 tablespoon of the vegetable oil in a skillet over medium heat. Stir in the onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Reduce heat to medium-low, and continue cooking and stirring until the onion is golden brown, 10 to 15 minutes more. Combine the noodles, onion, eggs, remaining vegetable oil, salt, and pepper in a large bowl. Pour mixture into an 8-inch square pan.
Bake in the preheated oven until firm, about 35 minutes.