For chicken: In a small bowl, combine cumin, salt, pepper and cayenne. Rub chicken breasts with spice mixture. Grill chicken 6 minutes on each side or until tender. Remove from heat and thinly slice; set aside.
For sautéed mushrooms: Heat butter in a sauté pan over medium heat. Add sliced mushrooms and onions. Season to taste with salt and pepper. Cook until mushrooms are soft. Remove from heat and set aside.
For basil sour cream: Combine basil, garlic, salt and pepper in a food processor. Process until pureed. Mix into sour cream. Set aside.
To assemble quesadillas: Pre-heat grill or griddle to 375 degrees Fahrenheit. Place tortillas on a flat surface. Generously sprinkle cheese onto one half of each tortilla. Combine chicken and mushrooms; divide evenly on top of the cheese. Sprinkle cheese on top of the chicken and mushroom mixture. Season to taste with salt and pepper.
Fold each tortilla in half (creating a half circle shaped quesadilla). Cook each quesadilla about 4-6 minutes, until cheese is melted and the tortilla begins to brown and crisp. Flip tortilla and repeat about 4-6 minutes. Remove quesadilla from heat and cool about 30 seconds before cutting into 3 triangle slices. Serve with basil sour cream.