Combine the chicken stock, soy sauce, oyster sauce, sake, sugar, sesame oil, and black pepper in a medium bowl and mix well.
Bring a large pot of water to a boil. Add the carrots, kale, and snow peas and cook for 2 minutes. Use a strainer to remove them from the water (keep the water boiling) and rinse well under cold water. Gently squeeze the kale to remove excess water. Cook the udon noodles in the boiling water, stirring occasoinally, just until the noodle clumps separate, about 2 minutes for fresh and 4 minutes for frozen. Drain well.
Heat the vegetable oil in a wide shallow pot or Dutch oven over medium-high heat until it shimmers. Add the mushrooms and scallions and cook, stirring occasionally, until the mushrooms start to wilt, about 2 minutes. Add the carrots, kale, snow peas, and seafood and cook, stirring occasionally, until the seafood is almost cooked through, about 2 minutes. Increase the heat to high, add the sauce, and bring to a boil. Add the noodles and cook, stirring, until the sauce thickens slightly and the noodles are well coated, about 2 minutes.
Divide among 4 plates and serve right away.