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Place the basil, arugula, nuts, cheese, garlic, and miso paste in the bowl of a food processor fitted with the blade attachment or blender. Process until the ingredients are finely chopped, about 30 seconds. Scrape down the sides of the bowl. With the motor running, stream in the olive oil and lemon juice, blending until the pesto is uniform; set aside.
Arrange the chicken in a single layer in a large saucepan. Add enough cold water to cover the chicken breasts by 1 inch. Bring to a boil over medium-high heat. Reduce the heat to low, cover the pan, and simmer until the chicken is opaque through the middle and an instant-read thermometer in the thickest part of the meat registers 165°F, 8 to 12 minutes depending on the thickness of the chicken breasts.
Transfer the chicken to a cutting board or large plate and use two forks to shred the meat while still warm. Cool completely.
Place the chicken, sun-dried tomatoes, mayonnaise, and pesto in a large bowl and stir until well-combined. Serve immediately in sandwich rolls with lettuce and tomato if desired or refrigerate.