Preheat the oven to 400 degrees F. Line a sheet pan with parchment paper.
Pat the tofu dry with a towel. Wrap it with a dry towel and place a cast iron pan or cutting board on top to remove more liquid. Let stand for 15 minutes. (You can use a tofu press if you have one.) Cut the tofu into 1-inch cubes.
Toss the tofu with 1 tablespoon oil and 1 tablespoon soy sauce in a large bowl. Sprinkle with the cornstarch, gently stirring until all the tofu cubes are coated.
Toss the broccoli with 1 tablespoon oil, 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper in a large bowl.
Arrange the tofu and broccoli in a single layer on the prepared sheet pan. Roast for 25-30 minutes until the tofu is lightly browned and the broccoli stems are crisp-tender, flipping over halfway through cooking.
While the broccoli and tofu are in the oven, make the peanut noodles. For the sauce, whisk together the peanut butter, garlic, ginger, soy sauce, maple syrup, lime juice, red pepper flakes and water in a small bowl until fully combined. The sauce should be thick but pourable consistency.
Cook the noodles according to package instructions. Transfer them to a large bowl and toss them with half the peanut sauce, adding more as desired. (Try to save some sauce to drizzle over the bowls.)
To assemble, spoon the tofu, broccoli and noodles into bowls and top with sliced scallions and chopped peanuts. Drizzle with additional sauce as desired.