In a small skillet over medium-low heat, toast the sesame seeds, stirring them often, until lightly browned, 8 to 10 minutes. Remove the skillet and cool. The seeds can be toasted up to 3 days ahead. Store airtight, at room temperature.
Bring a large pot of water to a boil with 2 tablespoons salt. Add the snow peas and cook, stirring once or twice, until crisp-tender, 1 to 2 minutes. With a slotted spoon, transfer the snow peas to a bowl of iced water. Cool completely, then drain immediately and pat dry. The snow peas can be prepared several hours in advance. Cover and hold at room temperature.
Meanwhile, return the same pot of water to a boil and add another tablespoon of salt. Add the noodles, partially cover and cook according to package directions. Drain, rinse thoroughly with cold water, and drain again. In a large bowl, toss the noodles with the sesame oil. The noodles can be prepared several hours in advance. Cover and hold at room temperature
In a large skillet over high heat, warm the canola oil. Add the shiitake and cremini mushrooms with ½ teaspoon of salt and cook, tossing and stirring occasionally, until barely tender, about 4 minutes. With a slotted spoon, transfer the mushrooms to the bowl with the noodles.
Just before serving, add the snow peas to the bowl with the noodles and mushrooms. Re-whisk the dressing and pour it over the salad. Toss well. Add the sprouts, green onion and 2 tablespoons of the sesame seeds and toss again. Turn the salad out onto a deep serving platter. Sprinkle the remaining seeds over the salad and serve immediately.