To prepare spring rolls, heat oil in a large nonstick skillet over medium-high heat. Add carrots, 1 1/2 cups red onion, green onions, mushroom, and 1 1/2 cups red bell pepper; sauté 3 minutes or until slightly tender. Remove from heat; stir in soy sauce and chile sauce. Cool 5 minutes. Stir in cheese.
Working with 1 egg roll wrapper at a time (cover remaining wrappers to prevent drying), spoon about 1/3 cup vegetable mixture into center of each wrapper. Fold lower right corner over vegetable mixture; fold lower left and top right corners over vegetable mixture. Moisten top left corner with water; roll up jelly-roll fashion. Coat spring rolls with cooking spray; place rolls, seam sides down, on a baking sheet. Bake at 425° for 18 minutes or until golden brown, turning rolls over halfway through baking time. Cut rolls crosswise at an angle; serve immediately with salad.