Bring a large pot of salted water to a boil. Add the broccoli and asparagus and cook for 3 minutes. Remove with a slotted spoon and place in an ice bath to stop the cooking. Add the pasta and cook until al dente (according to package instructions). Drain, saving 1 cup of pasta water. Toss the pasta with 1 tablespoon olive oil. Set aside.
In a large saucepan, heat 5 tbsp olive oil over medium heat. Add about ⅔ of the garlic (2 minced cloves) and cook until just starting to turn golden, about 2 minutes. Add the mushrooms and cook until soft and just starting to release liquid, about another 3 to 5 minutes.
Drain the broccoli and asparagus and add to the saucepan with the mushrooms. Add in the zucchini and peas and cook until soft, about 4 to 5 minutes.
Carefully stir in the cooked pasta, cream, Parmesan cheese, butter, 1 tsp salt, and pepper. Stir to fully incorporate. Taste and add another ½ tsp of salt, if desired. If you feel it needs it, add a little of the pasta water to loosen the sauce even more.
Meanwhile, heat a medium-small saucepan over medium heat. Add 1 tbsp olive oil and heat until shimmering. Add the remaining garlic (about 3 minced cloves), tomatoes, basil, and a pinch of salt and pepper. Saute until tomatoes are wilting and just starting to break down.
Pour the tomato mixture over the top of the pasta and garnish with toasted pine nuts. Serve at once.