Heat grill to medium (350° to 450°).
Mix soy sauce, mirin, and sesame oil in a bowl. Add tofu and marinate 10 minutes.
Heat a large pot of water to boiling. In a bowl, toss mushrooms in oil to coat.
Drain tofu, reserving marinade. Slip cubes onto 4 metal or soaked wooden skewers, alternating with green onions and mushrooms. Thinly slice any leftover onion.
Boil noodles in water until al dente, 4 minutes. Drain; rinse well with cold water. Divide noodles among dinner bowls and garnish with radishes and sliced onion.
Oil cooking grate, using tongs and a wad of oiled paper towels. Grill tofu skewers, covered, turning once, until lightly browned, 6 to 8 minutes. Set skewers on noodles.
Mix marinade with broth, wasabi, lemon zest, and juice and serve on the side.
*Buy nigari in the refrigerated section of your grocery store.
Note: Nutritional analysis is per skewer plus 1/4 of noodles.