Spread a thin layer of pesto on the cut sides of the rolls. Layer the bottom halves of each roll with a quarter of the chicken, spinach, and mozzarella and replace the tops.
Lightly coat the top and bottom of the panini with cooking spray. Heat a large nonstick skillet over medium-low heat. Place paninis in the pan, and place another heavy skillet on top of the sandwiches to press them down. Cook 2 to 3 minutes per side or until golden brown and cheese is melted.
Season 3/4 lb skinless boneless chicken thighs (3 thighs) with salt and pepper. Grill or broil until no longer pink (170°F), about 15 minutes. Refrigerate until needed.