Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente. Drain and set aside.
While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the minced garlic and cook for 30 seconds until fragrant.
Add the broccoli florets and cherry tomatoes to the skillet. Cook for 4 to 5 minutes, stirring occasionally, until the broccoli is tender but still bright.
Stir in the basil pesto and heavy cream. Let the mixture simmer gently for 2 minutes to warm through and combine.
Add the drained pasta and fresh spinach to the skillet. Toss everything together until the spinach wilts and the pasta is evenly coated.
Remove from heat, sprinkle with Parmesan cheese, and season with salt and pepper. Toss once more before serving.