Bring 4 quarts of water to a boil in a large pot for the noodles.
Meanwhile, heat 1 tablespoon of the oil in a large skillet over medium-high heat until shimmering. Add half of the cabbage, season with 1/4 teaspoon of salt, and cook, tossing frequently, until golden brown, 5 - 8 minutes. Transfer the cooked cabbage to a place. Repeat with the remaining cabbage.
Add 1 tablespoon of salt and the noodles to the boiling water and cook until al dente.l Drain the noodles and transfer back to the pot. Add the reserved cabbage and butter, toss to combine, and season to taste with salt and pepper. Serve.