Place noodles in a large bowl; cover with warm water. Let stand 20 minutes. Drain; set aside.
Combine the mushrooms and boiling water in a bowl; let stand for 20 minutes. Strain through a sieve into a bowl, reserving the mushroom liquid. Cut mushrooms into strips.
Heat peanut oil in a wok or nonstick Dutch oven over medium-high heat. Add onion, jalapeño, ginger, and garlic; stir-fry for 1 minute. Add mushrooms, carrot, and salt; stir-fry 2 minutes. Stir in 1/4 cup reserved mushroom liquid; cover and cook 3 minutes or until carrots are crisp-tender and liquid evaporates.
Add bok choy; stir-fry 1 minute. Stir in noodles, remaining mushroom liquid, soy sauce, and tofu; cook for 2 minutes.
Combine 3 tablespoons water and cornstarch; pour into pan. Bring to a boil; cook 2 minutes or until slightly thick. Drizzle with sesame oil.