Fill a large saucepan with 1 inch of water and fit with a steamer basket (or fill a large skillet with 1/2 inch of water). Bring the water to a boil.
Place the pot stickers in the basket (or skillet), cover, and steam for 4 minutes. Add the snap peas and carrots, cover, and steam until the pot stickers are cooked through and the vegetables are tender, 4 to 6 minutes.
Meanwhile, in a small bowl, combine the soy sauce and sesame oil. Divide the bean sprouts among bowls and top with the pot stickers and vegetables. Sprinkle with the peanuts and scallions. Serve with the sauce.