The same salty-sweet mixture that flavors the tofu also coats the noodles. Consider making a double batch of the sauce and refrigerating half for use on chicken, fish, or pork another night.
Preheat oven to 400 degrees. Gently press tofu between paper towels to remove excess water. Cut tofu in half lengthwise, then into 1/2-inch sticks.
Line 2 rimmed baking sheets with nonstick baking mats. Arrange mushrooms on 1 sheet and tofu on the other. Mix 1 tablespoon each of soy sauce, honey, and vinegar in a small bowl. Brush mixture on tofu, reserving leftover sauce. Bake mushrooms and tofu until golden, about 15 minutes for mushrooms and 18 minutes for tofu. (Flip tofu when you remove mushrooms from oven.) Let cool slightly on sheets, then combine mushrooms and tofu in a large bowl.
Add oils, orange juice, mustard, remaining soy sauce, honey, and vinegar to reserved sauce. Toss half with tofu and mushrooms.
Using a vegetable peeler, shave cucumbers into long ribbons, turning after each stroke. Stop at core, and discard. Transfer ribbons to a bowl.
Cook udon in a large pot of boiling water, following package directions. Drain, and toss with cucumbers and remaining sauce. Divide among bowls, and top with tofu and mushroom mixture and sauce. Garnish with sprouts, and serve immediately.