Cook Mozzarella Sticks according to package directions. Cut each in half lengthwise.
Meanwhile, heat oil in large skillet set over medium-high heat; cook mushrooms, salt and pepper for 8 to 10 minutes or until golden brown and tender. Transfer to plate and wipe out skillet with paper towel.
Divide mushrooms, Mozzarella Sticks and cheddar between 2 tortillas. Top with remaining tortillas.
Heat skillet over medium-low heat; cook quesadillas, one at a time, for 3 to 5 minutes per side or until tortillas are golden brown. Cut into wedges and serve with salsa.