Tear off four 15x12-inch pieces of parchment paper. Halve or quarter any large mushrooms. Cut each chicken breast portion in half horizontally.
Place a chicken portion on half of each sheet of parchment paper. Top with mushrooms, sweet pepper, spinach, dried tomatoes, and pesto. Sprinkle with salt and black pepper. For each packet, fold parchment paper crosswise over filling. Fold and crimp edges to tightly seal in a half-moon shape.
Arrange packets, two at a time, on a microwave-safe plate. Microwave on 100% power (high) for 6 to 8 minutes or until chicken is no longer pink (165 degrees F). Repeat with the remaining packets. Let stand for 3 minutes before carefully opening packets. If desired, garnish with fresh basil.