Place the chicken breasts into the dish of a slow cooker. Pour the Buffalo sauce over the chicken, and stir to combine. Cover and cook on low for 7 to 8 hours or high for 4 to 5 hours. If you have rottiserie chicken, jump to the next step.
Once the chicken is cooked through, shred the meat with two forks and toss to coat with the sauce.
Assemble:
Place the tortillas on a flat surface. Top 4 tortillas with the chicken, Cheddar cheese, and blue cheese. Assemble the quesadillas.
Grill:
In a large skillet, melt butter over medium-high heat. One at a time, place the quesadillas into the skillet and grill for 3 minutes or until the cheese begins to melt. Flip and grill for another 3 minutes or until the cheese is completely melted. Remove from the skillet and repeat with remaining quesadillas.
Slice and serve with Ranch or Blue cheese dressing.