In a large skillet, heat the vegetable oil until shimmering. Add the mushrooms, season with salt and pepper and stir-fry over high heat until browned, about 8 minutes.
In a large pot of boiling salted water, cook the udon until al dente; drain. Return the udon to the pot. Add the chicken stock, tofu, soy sauce, kimchi and stir-fried mushrooms and bring to a simmer over moderate heat. Add the scallions and sesame oil and serve right away.