Marinate the chicken thighs – Add the trimmed chicken thighs to a gallon size zip top bag. Pour in the juice from the can of pickled jalapeños. Add lime juice, onion slices, about 1 tablespoon of oil, and 2 tablespoons Mexican seasoning. Work the marinade around so the chicken is well-coated. Zip the bag up, removing any air from the bag, and marinate in the refrigerator for 2-4 hours.
Heat griddle to medium-low to medium heat, about 375-400℉. Add about ½ tablespoon cooking oil and spread it into a thin layer with your spatula.
Mix the quesadilla-style cheese and pepper jack cheese together in a bowl.
Cook the Mexican chicken – Place the chicken thighs top side (smooth side) down on the oiled griddle. Cook untouched about 5-7 minutes on the first side, or until a nice crust develops. Then flip to the second side and cook an additional 5-7 minutes. After flipping, add the onions from the marinade to the chicken to cook as well. Continue to cook, flipping a few times so the chicken cooks evenly, until the internal temperature reaches 165℉. Then transfer chicken and onions to a cutting board to rest.
Cook the fajita vegetables - Scrape the griddle surface clean. Then add about ½ tablespoon cooking oil and spread it around with your spatula. Add the sliced onions and bell peppers to the griddle. Season with salt and pepper. Sauté over medium until they’re just soft. When peppers and onions are almost done to your liking, add the sliced tomatoes to cook briefly. Then transfer veggies to a bowl.
Cut chicken - After it's done resting, and while vegetables are cooking, cut the chicken into small bite-sized cubes.
Scrape down the griddle to remove excess food residue to prepare for cooking the quesadillas.
Build the chicken quesadilla - Add about ½ tablespoon butter to the heated griddle, and spread it around to the size of your tortilla. Lay your flour tortilla out flat on the buttered griddle. Sprinkle about ½ - ¾ cup cheese evenly across the whole tortilla, and top with about ⅕ of the chicken and sautéed veggies. Cook laid open and flat until the cheese starts to melt, about 1-2 minutes. Repeat for all the quesadillas.
Fold the quesadilla and cook until done - After a few minutes, fold the tortillas in half to form quesadillas. Cook on both sides until crispy and golden brown. Then transfer to a cutting board to cool briefly.
Slice and serve - Slice each quesadilla into 3-4 triangles. Serve immediately with sour cream, salsa, or your favorite quesadilla toppings.