Asparagus and garlic are the VIPs (very important prebiotics) in this dish. Serve kimchi on the side for a nice hit of probiotics if your kids are still building up their spice tolerance.
Bring a large saucepan of water to a boil over high. Remove from heat. Submerge noodles in water until tender, about 3 minutes. Drain; rinse with cold water.
Preheat oven to 450°F with racks in upper- and lower-third positions. Toss together mushrooms, asparagus, garlic, and 2 Tbs. of the sesame oil on a large rimmed baking sheet. Toss together tofu and remaining 1 Tbs. sesame oil on a separate large rimmed baking sheet. Place mushroom mixture on upper rack and tofu on lower rack of preheated oven. Roast mushroom mixture, stirring twice, until tender, about 15 minutes. Remove from oven; set aside. Increase oven temperature to broil. Stir tofu, and move to upper rack. Broil until lightly browned, about 4 minutes.
While mushroom mixture and tofu cook, microwave vinegar in a microwave-safe bowl until steaming, about 1 minute. Stir in carrots; let stand until cool, about 10 minutes.
Stir soy sauce, hoisin, and 2 Tbs. vinegar from pickled carrots in a separate bowl. Drain carrots.
Divide noodles among four bowls. Top with mushroom mixture, tofu, soy-sauce mixture, carrots, and kimchi, if desired.