In a large skillet, melt butter with olive oil over medium heat. Add minced garlic and diced onion, sauté until translucent, about 5-7 minutes. Season with 1 teaspoon of salt.

Stir in chopped broccoli and sauté for another 5 minutes, until the broccoli is tender yet still vibrant green. Add another teaspoon of salt.

Pour in the heavy cream. Bring the mixture to a gentle simmer, then add the shredded mozzarella cheese. Stir continuously until the cheese has melted and the sauce is smooth. Season with salt and pepper to taste.

Meanwhile, cook the pasta in a separate pot according to package instructions until al dente. Reserve ½ cup of pasta water, then drain the pasta.

Add the cooked pasta to the skillet with the broccoli and creamy sauce. Pour in the reserved pasta water and toss everything together gently until the pasta is well coated with the sauce.

Remove the skillet from the heat. Sprinkle grated parmesan cheese over the pasta and stir until the cheese is melted and incorporated.