Place a large skillet or grill pan over medium heat and drizzle in the olive oil. While the pan warms, heat your tortillas in the microwave for 15 seconds under a damp paper towel to ensure they stay pliable.

Stir together the shredded chicken, mozzarella, cheddar, cream cheese, garlic, and parsley in a large mixing bowl. Season with salt and pepper, adding the optional spinach now if you want an extra boost of greens.

Lay the tortillas flat and distribute the chicken and cheese mixture evenly across the center of each wrap, using about half a cup per tortilla. Fold the sides in and roll them up tightly, ensuring the ends are tucked so the filling stays inside.

Lay the wraps into the hot skillet with the seam side facing down to seal them shut. Grill for about 3 minutes on each side, allowing the exterior to become crisp and golden while the cheese melts into a gooey center.

Take the wraps off the heat once they feel firm and the cheese begins to peek out of the ends. Use a sharp knife to slice them on a diagonal and serve while the cheese is still hot and stretchy.