Bring a 5- to 6-quart pot of water to a boil.
In a 10- to 12-inch sauté pan, warm canola oil over medium-high heat. Add shallot, carrot, garlic, and ginger and sauté for 1 minute. Stir in mushrooms, reduce heat to low, and cover pan; sweat mushrooms until soft, about 4 minutes. Uncover pan and increase heat to medium-high again. Stir in edamame and sauté until heated through, about 2 minutes. Stir together broth, soy sauce, and lemon zest and pour into pan. Bring to a boil.
Stir in spinach a handful at a time, stirring after each addition until wilted. Stir in tofu, then turn off heat under pan. Season to taste with pepper.
Drop soba noodles into boiling water; cook until al dente, about 5 minutes. Drain in a colander and rinse with cool water to remove excess starch. Add noodles to sauté pan and turn on heat to medium-high. With tongs, toss noodles with vegetable mixture just until heated through, then divide among shallow pasta bowls and serve.