How to Make It
In a medium bowl, whisk the mayonnaise with the pesto. Add the chicken and mix well. Season with salt and pepper.
Preheat a cast-iron griddle or grill pan. Brush both sides of each piece of bread with oil. Spoon the chicken salad onto the bottoms and top with the spinach and cheese. Close the sandwiches and griddle or grill over moderate heat, turning once, until toasted and the cheese is melted, about 7 minutes. Cut the panini in half and serve hot.