For the vegetables: Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the carrots and cook for 1 minute. Then add the broccoli; cook for another minute. Add the red peppers; cook 1 minute more. Remove the vegetables from the skillet and set aside.
To the skillet, add the butter and the remaining 1 tablespoon oil; allow to heat up. Add the mushrooms, zucchini and squash, and cook until starting to soften, 2 to 3 minutes. Remove the vegetables from the skillet and add them to the other vegetables. Set aside.
For the sauce: Add the garlic and onions to the skillet, and cook until starting to turn translucent, about 2 minutes.
Next, pour in the chicken broth and wine; stir, scraping the bottom of the pan to loosen all the flavorful bits. Cook for 3 to 4 minutes, until reduced by about half. Stir in the cream and Parmesan, and allow the cheese to melt. Add salt and black pepper to taste.
Add the cooked vegetables to the sauce, along with the peas and basil; stir to combine. If the sauce needs a little more liquid, splash in a small amount of broth. Place the cooked pasta in a large serving bowl and pour all the contents of the skillet over the pasta; toss to combine. Sprinkle with extra basil, and serve with extra Parmesan.
If you're averse to wine, use all chicken broth instead.