Vegan Zha Jiang Mian

Name: Vegan Zha Jiang Mian
Ingredients: 1 tablespoon cornstarch; 1/4 cup vegetable oil; 1 medium Chinese eggplant (about 8 ounces), cut into 1/2-inch pieces (about 2 cups); Kosher salt; 1 small onion, finely chopped; 2 ounces fresh shiitake mushrooms, stemmed and cut into 1/4- to 1/2-inch pieces (about 1 cup); 1/2 block firm tofu (7 to 8 ounces), drained; 1/4 cup Chinese soybean paste; 3 tablespoons sweet bean paste (see Tip); 2 garlic cloves, finely chopped; 12 ounces dried udon noodles; Mung bean sprouts and shredded cucumber, for serving