In a blender, place basil, garlic, pine nuts, and parmesan cheese, pulse while adding olive oil. Once incorporated, remove from blender and set aside.
Turn on half the barbeque to medium or 350 F (175 C). In a bowl, toss the potatoes in garlic butter. Put the potato mixture in a tinfoil basket. Place the basket on the unlit side of the barbeque, using indirect heat, with the lid closed, and bake for 1 hour, topping with cheddar cheese halfway through the cooking process.
Remove the tender from the chicken breasts, toss the breasts in olive oil, salt, and pepper. Heat the barbeque to high 400 F (205 C). Oil the grill. Place chicken top breast side down on the grill, cook for 30 seconds then turn the breasts, cook for another 30 seconds more to create a crosshatch pattern, turn the chicken over and repeat the process.
Put chicken aside for 5 minutes to cool. Once cooled, make 3 cuts across the top of the chicken breasts about ½-inch apart, making sure not to cut completely through. In each cut, place first the spinach, then the swiss cheese, and finally the sun-dried tomato.
Turn on half of the grill to medium or 350 F (175 C). Place the breast on the unlit side of the grill, cook for 15 minutes with the lid closed or until an internal temperature of 165 F (74 C). Remove to a plate. Serve with potato and the pesto.
