Heat a panini press (or close-top grill) to medium heat. Arrange bell peppers and onions on the plate and close. Grill for 4-5 minutes.
Add the chicken strips to the grill. Pour the Italian dressing over top and close the top. Grill until the chicken is warmed through and the peppers and onions are soft. Remove from grill.
Working with one tortilla at a time, lay one tortilla on the grill plate. Sprinkle with 3-4 tablespoons shredded cheese, then top with a second tortilla. Close the grill and cook for 1-2 minutes, until the cheese is melted.
Arrange half of your cooked chicken, peppers and onions on one side of the tortilla and sprinkle with cilantro, more shredded cheese (1/4 cup) and cojita cheese (2 tablespoons). Use a spatula to fold over the tortilla. Close the grill and cook until the cheese is melted and the tortilla is crisp.
Remove from grill and cut into wedges. Repeat with remaining ingredients.