Preheat the oven to 350 degrees F.
Pour the water into a shallow baking dish and place the squash in the water face down.
Cover with foil and bake for 45 to 50 minutes, or until the squash is tender.
Meanwhile, lightly toast the walnuts in a dry sauté pan over medium-low heat for about 5 minutes.
Combine the walnuts, basil, garlic, ½ cup of the olive oil, and salt in a food processor and blend until smooth.
Set aside.
In a large skillet over medium heat, add the remaining 1 tablespoon olive oil and mushrooms.
Cover and cook until the mushrooms are tender, 2 to 3 minutes.
Add the broccoli, carrots, and broth and sauté for 5 to 7 minutes, or until broccoli is just starting to get tender (do not overcook).
Reduce the heat to low and add the peas and spinach.
Use a fork to scrape the squash flesh directly into the skillet.
Add the pesto and stir well to combine all ingredients.
Season to taste with pepper.