Spread pesto on one side of each bread slice. On two slices, layer the mozzarella, chicken, tomato slices, and spinach. Season with salt and pepper.
Top with remaining bread slices, pesto-side down, creating two sandwiches. Brush the outside of each sandwich with olive oil.
Heat a large skillet or panini press over medium heat. If using a skillet, place sandwiches in the pan and use another heavy pan to press down on top.
Cook for 5-6 minutes until golden and crispy, then flip and cook another 5-6 minutes, pressing occasionally.
Remove when the cheese is melted and the bread is beautifully toasted. Let cool for 2 minutes, slice diagonally, and serve while hot!