Preheat oven to 400 degrees F. Coat a large rimmed baking sheet with cooking spray.
Using a fork, scrape the squash from the shells into a large bowl. Add the chicken and toss gently to combine. Serve topped with the pesto.
Place halved lengthwise and seeded squash, cut-side down, on one side of the prepared pan. Bake until tender, about 45 minutes.
Meanwhile, combine avocado, basil leaves, pistachios, lemon juice, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper in a food processor. Pulse until finely chopped. Add 4 tablespoons oil and process until smooth.
Ten minutes before the squash is done, toss trimmed and cut into 1-inch pieces chicken, the remaining 1 tablespoon oil and the remaining 1/4 teaspoon each salt and pepper together in a medium bowl. Spread the chicken in an even layer on the empty side of the baking sheet. Return to the oven and bake until just cooked through, about 10 minutes.
Percentages are based on a diet of 2000 calories a day.