Toss chicken with chili-powder, cumin, pepper and salt (about ¾ tsp salt and ½ tsp pepper).
Heat 2 teaspoon olive-oil in 12" nonstick skillet over medium-high heat. Add chicken & cook then turn occasionally until completely cooked through, about 6minutes. Transfer to a medium mixing-bowl.
Heat remaining 2 teaspoon oil in same skillet. Add bell-pepper, red-onion and simmer until tender about 6minutes. While incorporating in garlic throughout last a minute. Transfer to bowl with chicken, pour in lime & toss.
Brush top side of tortillas with melted butter. Wipe skillet clean with paper-towels, heat over moderate heat. Toss sweet together.
Transfer one tortilla down side to skillet. Working fast scatter ½ cup cheese evenly over a half of the tortilla then sprinkle with roughly ¾ cup of the chicken mixture. Twist top over, repeat with another tortilla.
Let cook until golden-brown on underside about two minutes, then carefully turn to reverse side and then continue to cook till golden-brown on underside.
Repeat procedure with remaining tortillas, cheese and filling. Cut each quesadilla into two or three wedges. Serve warm with desired toppings.