Separate chunks of chicken from tikka masala sauce. Roughly chop chicken and set aside. Spread approximately half (¼ cup or so) of tikka masala sauce on one side of one tortilla.
Heat large skillet over medium heat. Add ½ tablespoon butter to skillet and melt completely. Once butter is melted, place tortilla with sauce in skillet, sauce-side up.
Spread approximately ½ cup chopped chicken across top of tortilla, then cover with 1 cup shredded cheese. Place second tortilla on top, covering chicken and cheese completely.
Let quesadilla cook approximately 3 minutes, until cheese has melted to top tortilla. Carefully remove quesadilla from skillet with spatula or slide quesadilla out of skillet onto plate (see Notes).
Melt another ½ tablespoon butter in skillet. Once melted, flip quesadilla over and return to skillet to brown other side.
Cook 2 to 3 minutes, then transfer fully cooked quesadilla to plate. Repeat process with remaining 2 tortillas, to create second quesadilla.
Once both quesadillas are prepared, cut each quesadilla into quarters and serve with additional tikka masala sauce, chutney, and chopped cilantro if desired.