Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
Toss the butternut squash cubes with 1 tablespoon olive oil, ½ teaspoon salt, and ¼ teaspoon pepper until evenly coated.
Spread the squash in a single layer on the prepared baking sheet and roast for 25 minutes, flipping halfway through, until golden and tender.
While the squash roasts, bring a large pot of salted water to a boil and cook the penne pasta according to package directions until al dente. Reserve ¼ cup of pasta water before draining.
In a large skillet, heat 1 tablespoon olive oil over medium heat.
Add the chicken pieces, season with ½ teaspoon salt, ¼ teaspoon pepper, and Italian seasoning, and cook for 6–8 minutes until golden and cooked through.
Turn the heat to low and add the cooked pasta, roasted squash, and pesto to the skillet with the chicken.
Pour in a splash of the reserved pasta water to help coat everything evenly.
Toss gently until all ingredients are well combined and the pasta is glossy.
Stir in ⅓ cup Parmesan cheese until melted and incorporated.
Remove from heat and taste, adjusting seasoning with a pinch of salt or pepper if needed.
Transfer to serving bowls or a large dish.
Top with chopped walnuts, extra Parmesan, and fresh basil leaves before serving.