Preheat the panini grill pan over medium heat for 3 - 4 minutes. Brush the grill with oil and pan roast the chilies for 2 - 3 minutes or until they are charred. Remove from the heat. Seed chilies, if desired, and chop coarsely.
Combine the chilies and the remaining salsa verde ingredients in the bowl of a food processor fitted with a metal blade. Process the mixture until puréed.
Place 2 tortillas on a work surface. Divide the queso fresco and colby cheeses evenly between the tortillas. Layer the chicken on top of the cheese, followed by the tomato slices, cilantro, 3 - 4 Tbsp. salsa verde and the remaining 2 tortillas.
Press the quesadillas firmly together. Brush one side of one quesadilla with oil.
Preheat a panini grill pan over medium heat for 3 - 4 minutes. Place one quesadilla, oiled side down on the hot pan. Place the panini press on top of the quesadilla and cook for 1- 2 minutes, or until golden and crispy.
Brush the top side of the tortilla with oil and carefully flip it over. Cover with the panini press and cook the reverse sides for 1 - 2 minutes.
Transfer the quesadilla to a cutting board and keep warm while the second quesadilla is cooking.
Repeat the process with the remaining quesadilla. Use a sharp chef's knife to cut each quesadilla into quarters and serve while hot. Serve with salsa verde.