Preheat the oven to 425 degrees F. Melt the butter in a large ovenproof skillet over medium-high heat; remove 1 tablespoon to a small bowl. Add the carrots, celery and onion to the skillet; cover and cook, stirring occasionally, until slightly softened, about 5 minutes. Add the mushrooms and soy sauce; cover and continue cooking, stirring occasionally, until tender, about 3 minutes. Add the flour; cook, stirring, 1 minute.
Add the milk, 1 cup water and 1/2 teaspoon salt and stir until the flour is incorporated. Bring to a boil, then reduce the heat to medium and simmer, stirring occasionally, until thickened, about 5 minutes. Add the tofu and frozen peas; return to a simmer. Season with salt and pepper. Remove from the heat.
Brush the bread with the reserved melted butter and cut into quarters. Arrange buttered-side up on the tofu mixture. Transfer the skillet to the oven and bake until the bread is toasted, about 8 minutes.