Cook the soba noodles according to the package directions. Drain and set aside.
Heat the oil in a large cast iron pan or heavy-bottomed skillet over medium heat. Add onion, ginger, and garlic, and cook until the onions are soft and ginger is aromatic, about 5 minutes. Add the mushrooms to pan and stir. Cook, stirring occasionally, until mushrooms begin to soften, about 3 minutes. Add the carrots. Cook, stirring occasionally, until the carrots are slightly tender, about 3 minutes more.
Whisk together the soy sauce, sesame oil, peanut butter, and vinegar in a small bowl. Pour the mixture into the skillet and stir into the mushrooms and carrots. Add the reserved noodles. Raise the heat to high and cook, stirring frequently, until noodles and vegetables are warmed through and well combined, 1 minute. Add the scallions and stir. Remove from the heat. Divide the noodles evenly between two plates. Drizzle each serving with a teaspoon of lime juice and garnish with sesame seeds.