Fill a large pot with lightly salted water and bring to a boil. Add soba noodles; cook, stirring occasionally, until tender yet firm to the bite, 3 to 5 minutes. Drain and toss with 1 tablespoon oil. Allow to cool, about 15 minutes. Transfer to a bowl.

Preheat oven to 375 degrees F (190 degrees C).

Heat 2 tablespoons oil in a large skillet over medium heat. Add onion and scallions; cook and stir until onion is translucent, about 5 minutes. Stir onion mixture into the noodles in the bowl. Add eggs, salt, and black pepper; mix until well-combined.

Pour remaining oil into a 9-inch baking dish. Place dish in the oven to heat the oil, about 2 minutes. Remove from the oven and pour in noodle mixture.

Bake kugel in the preheated oven until golden, about 45 minutes.