Corn, shiitakes, and aged cheese bring meaty umami flavor to this dish. If you can't find fresh Anaheim chiles, substitute about 1/4 cup chopped canned green chiles. Sharp cheddar or aged Gruyère cheese could also stand in for the Gouda, if you prefer.
Cut chiles in half lengthwise; discard seeds and membranes. Place chile halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil 8 minutes or until blackened. Place in a zip-top plastic bag; seal. Let stand 15 minutes. Peel and chop. Reduce oven temperature to 200°.
Heat oil in a large nonstick skillet over medium-high heat. Add mushrooms; sauté 2 minutes. Add corn, onions, and pepper; sauté 2 minutes. Place mixture in a bowl; stir in chopped chiles. Wipe pan clean.
Place about 1/4 cup mushroom mixture and 1/4 cup cheese over half of 1 tortilla. Repeat procedure with remaining 1 1/4 cups mushroom mixture, remaining 3/4 cup cheese, and remaining 3 tortillas. Heat pan over medium heat. Coat pan with cooking spray. Place 1 tortilla in pan; cook 2 minutes or until cheese melts and bottom is golden. Fold tortilla in half; place on a baking sheet. Place in 200° oven to keep warm. Repeat procedure with remaining tortillas. Cut each quesadilla into wedges; serve with salsa.