Mince garlic and add to a small bowl. Add soy sauce, water, sesame oil, brown sugar (2 tablespoons of honey may be substituted here) and cornstarch; whisk together and set aside.
Bring a medium pot of water to boil and add the broccoli. Boil for 5 minutes, then add to a large bowl of ice water to stop the cooking process. Set aside.
Spray a medium, non-stick skillet with cooking spray; add chicken and heat over medium-high. Brown on both sides, cooking until no longer pink in the center; chop or shred as desired.
Drain broccoli and add to the skillet with the chicken; turn heat down to medium-low. Gently pour in the garlic mixture and heat about 5 minutes until thickened. Toss together until everything is well coated and heated through; remove from heat. (If mixture seems to thick or sticky, add 1-2 tablespoons of water for a thinner consistency).
Heat a separate sauté pan over medium-high heat.
Place 1/4 cup of the chicken mixture on one half of a tortilla. Sprinkle grated cheese on top of the chicken, then fold the tortilla in half to cover the chicken and cheese.
Cook quesadilla on both sides until cheese is melted and tortilla is slightly browned. Repeat with remaining ingredients.
Cut each quesadilla into triangles and serve.