Bring 3 quarts of water to a boil in a large pot. Add soba noodles; cook, stirring occasionally until tender, 4 to 6 minutes. Drain. Set aside.
Heat vegetable oil and sesame oil together in a 10-inch skillet or wok over medium heat until shimmering, 2 to 3 minutes. Place chicken in skillet and cook until no longer pink in the center and juices run clear, 7 to 10 minutes. Add garlic; cook until fragrant and softened, 1 to 2 minutes. Mix in mushrooms and carrots and cook until softened, 2 to 3 minutes. Stir in spinach; cover, and cook until spinach wilts, 3 to 4 minutes.
Add cooked soba noodles to vegetable mixture and toss to combine. Serve hot.