Soak the mushrooms: Soak the dried mushrooms in 2 cups of room temperature water for 30 minutes, while you complete the steps below.
Prep the veggies: Cut the tofu into 2 x 2 inch squares and pat dry. Slice the bok choy in half, if using. Slice the onion. Thinly slice the garlic. Peel and grate the ginger. Remove the stems from the shiitake mushrooms and thinly slice them. Thinly slice the sweet peppers. (For the garnish, thinly slice the radishes and set aside; you can also do this later if you prefer.)
Saute the bok choy and tofu: In a large skillet, heat 1 tablespoon sesame oil and 1 tablespoon olive oil over medium high heat. If using bok choy, add the bok choy cut side down and cook until lightly charred, about 3 minutes (if using spinach, skip this step). Set aside. Add the tofu to the same pan and cook for about 10 minutes until lightly browned on all sides, turning occasionally. Remove and place on a plate; drizzle lightly with soy sauce.
Cook the noodles (while you make the broth): Heat a large pot of water to a boil and cook noodles to al dente while you make the broth (do not salt the water). Drain and set aside. If necessary, refresh the noodles under some hot tap water before serving.
Make the broth: In large pot or Dutch oven, heat 1 tablespoon sesame oil and 1 tablespoon olive oil over medium high heat. Add the onion and cook 4 minutes until lightly browned on the edges. Add the garlic and ginger and saute for 30 seconds just browned, stirring constantly. Add the vegetable broth and bring to a simmer.
Add the dried mushrooms to the broth: Use your hands to remove the soaked dried mushrooms from the bowl, saving the soaking liquid. Rinse them and add them to the broth. Pour the remaining soaking liquid through a fine-mesh strainer into the broth (this avoids adding any tough particles).
Add the veggies: Immediately add the soy sauce, mirin, and the shiitake mushrooms to the broth. Cook 8 minutes. Add the peppers and bok choy (or spinach) and cook for 1 minute. Taste the broth and season with salt until the flavor pops.
Serve: To serve, place the noodles four bowls and top with broth, tofu, and vegetables. Garnish with sliced radishes and garlic chili sauce. Store leftovers with the broth and noodles in separate containers (to avoid the noodles soaking up all of the broth).