Bring a large pot of salted water to a boil. Fill a bowl with ice water. Add the broccoli to the boiling water and blanch for 1 minute. Remove to the ice water. Allow to cool, then drain and dry on paper towels. Cover the pot of water and keep hot.
Heat a wide, heavy skillet or wok over medium-high heat and add 1 tablespoon of the oil. When it is rippling hot, add the mushrooms and sear for 5 minutes, stirring from time to time. Turn the heat to medium, add the garlic, ginger, scallions, and red pepper flakes and cook for another 30 seconds to a minute, until fragrant. Remove from the heat and transfer the contents of the pan to a bowl.
Return the pan to medium-high heat and heat the remaining tablespoon of oil until rippling. Add the tofu and sear on both sides until it begins to color, about 1 minute per side. Add 1 teaspoon soy sauce and stir together. Return the mushroom mixture and the broccoli to the pan. Add the stock and soy sauce, bring to a simmer and turn the heat to low. Simmer uncovered for a minute or two, until the broccoli is crisp-tender, then turn off the heat.
Meanwhile, bring the pot of water back to a boil and add the pasta. Cook soba or udon noodles until al dente, about 5 minutes. Cook softened rice noodles for 1 minute. Drain and toss with the mushrooms and the broccoli. Heat through, add the cilantro and sesame oil, toss together, and serve.