Preheat a panini press or a large skillet over medium heat.
Spread the basil pesto evenly on the inside of all four bread slices.
Layer the mozzarella cheese, sliced chicken, roasted red peppers, red onion, cherry tomatoes, fresh basil, and arugula on the bottom two bread slices.
Top with the remaining bread slices to close the sandwiches.
Brush the outside of the sandwiches lightly with olive oil or butter.
Grill in the panini press for 5 to 7 minutes, or in a skillet for 3 to 4 minutes per side using a heavy pan to press down, until the bread is golden and the cheese is melted.
Remove from heat, carefully slice in half, and serve immediately.