Bring a large pot of well-salted water to a boil over high heat. Add the pasta and cook until barely al dente, about 1 minute less than package timing. Add the green peas and cook for about 30 seconds. Add the snow peas, stir, and immediately drain the vegetables and pasta in a colander set in the sink. Rinse with cool water to stop the cooking. Drain well, toss with 1 Tbs. of the vegetable oil, and set aside.
Heat 1 Tbs. of the vegetable oil in a 12-inch skillet over medium heat. Add the onion, shiitake, garlic, ginger, 1/2 tsp. salt, and a few grinds of pepper. Cook, stirring occasionally, until the onions are opaque and the mushrooms have released their juices, 3 to 4 minutes—don’t let the vegetables brown. Remove the pan from the heat, transfer the vegetables and any juices to a small bowl, and let cool to room temperature.
In another small bowl, whisk the remaining 1 Tbs. vegetable oil with the soy sauce, vinegar, sesame oil, and sugar.
In a large bowl, combine the cooled pasta and vegetables, scallions, and 1 Tbs. of the sesame seeds. Toss with the dressing and season to taste with salt and pepper. Serve at room temperature, garnished with the remaining sesame seeds.
You can prepare the salad up to 2 hours ahead.