Heat 1 tablespoon of the olive oil in a skillet over medium-high heat. Chop or dice the chicken into bite-sized pieces. Sprinkle the chicken with salt, pepper, and taco seasoning. Add the chicken to the skillet and sauté until browned. Because the pieces are small, the chicken will be cooked through. Set aside.
Add the remaining 1 tablespoon olive oil to the skillet over medium-high heat. Slice onions and peppers, and add them to the skillet. Cook until the peppers are tender, 3 to 4 minutes. Remove and set aside.
Turn down the heat slightly to medium-low and melt 1/2 tablespoon of the butter in the skillet or griddle. Lay a flour tortilla in the skillet. Add grated cheese on the bottom tortilla, and then add the chicken and cooked peppers/onion mixture. Top with more grated cheese and cover with another tortilla.
When the tortilla is golden on the first side, carefully flip the quesadilla to the other side, adding another 1/2 tablespoon butter to the skillet at the same time. Continue cooking until the second side is golden. Repeat to make additional quesadillas.
Cut each quesadilla into wedges and serve with pico de gallo.