Asian Noodles in Broth with Vegetables and Tofu

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One 8-ounce package udon or soba noodles

3 cups water

1/2 cup soy sauce, preferably naturally brewed

1/3 cup mirin

3 tablespoons rice vinegar

1 tablespoon finely grated peeled fresh ginger

1 cup snow peas, strings removed and halved crosswise on the diagonal

1 cup thinly sliced mushrooms

16 ounces soft tofu, cubed

1 romaine lettuce heart, shredded

1/2 cup thinly sliced red radishes or daikon matchsticks

1/2 cup carrot matchsticks

2 scallions, white and light-green parts only, thinly sliced

Toasted sesame oil, for serving, optional

Hot sauce, for serving, optional