Heat the olive oil: In a large pan, heat 3 tablespoons of extra virgin olive oil over medium heat. Add the 4 crushed garlic cloves and sauté for about 1 minute. To prevent the garlic from burning, splash in a tablespoon of pasta water.
Sauté the mushrooms: Toss in the 4 oz of sliced mushrooms and sauté for 10-15 minutes. Season with salt and pepper as you cook, allowing the mushrooms to soften and develop a golden-brown color.
Cook the pasta: In a separate large pot, add water mixed with rock salt and bring it to a boil. Add 1 package of dried pasta and cook according to the package instructions, typically 9-12 minutes until al dente.
Combine pasta and mushrooms: Once the pasta is cooked, reserve a cup of pasta water and drain the rest. Add 1 tablespoon of butter to the sautéed mushrooms. Use tongs to transfer the drained pasta directly into the mushroom pan, mixing well. Incorporate 1/2 cup of freshly grated Parmigiano Reggiano and a bit of the reserved pasta water.
Add cream: Stir in 4 tablespoons of heavy cream, mixing everything until well combined—a few extra tablespoons can be added if a richer sauce is desired.
Final touch: Finish the dish by sprinkling 1 teaspoon of dried chili flakes and 1 teaspoon of dried parsley on top. Serve hot, and feel free to add more cheese if desired!