Combine the soy sauce, sesame oil, 1 teaspoon honey, garlic, ginger and 2 tablespoons canola oil in a baking dish. Add the mushrooms, turn to coat and let them marinate at room temperature for 15 minutes.
Brush the zucchini with 2 tablespoons of canola oil and season them with salt and black pepper, to taste. Grill until slightly charred on both sides and just cooked through, about 3 minutes per side. Remove and cut them crosswise into 1/4-inch slices. At the same time, grill the mushrooms until golden brown on both sides and just cooked through, about 4 minutes per side. Remove and cut them into thin strips.
Whisk together the rice vinegar, lime juice, mustard, 1 tablespoon honey and 1/4 cup canola oil in a large shallow bowl and adjust the seasonings with salt and pepper, to taste. Add the warm, cooked rice and toss to coat. Arrange the mushrooms, zucchini, bean sprouts and carrots over the top and garnish with cilantro leaves.