Bring a large pot of water to a boil, salt generously, and add the linguine. Cook until al dente according to package directions, then drain, reserving 1/2 cup of the cooking liquid.
Return the hot, drained linguine to the pot off the heat.
Add the butter, heavy cream, finely minced garlic, a pinch of salt, and some black pepper to the pasta; stir until the butter begins to melt.
Stir in 3/4 cup freshly grated Parmigiano-Reggiano until it melts and starts to coat the pasta.
Add reserved pasta cooking liquid a few tablespoons at a time (about 2–3 tablespoons to start) and toss until the sauce reaches your desired consistency; use up to the reserved 1/2 cup as needed.
Taste and adjust seasoning with more salt and pepper if needed, then transfer to a serving platter and garnish with chopped fresh parsley and extra grated cheese if desired.