Bring a large pot of water to a rolling boil, add a generous pinch of salt, and cook the linguine until al dente according to package directions.
Reserve 1/2 cup of the pasta cooking liquid, then drain the pasta in a colander.
Return the hot drained linguine to the empty pot (off the heat). Add the butter, heavy cream, minced garlic, a pinch of salt, pepper, and 3/4 cup grated Parmigiano-Reggiano.
Toss and stir until the butter and cheese begin to melt and coat the pasta, adding reserved pasta water a few tablespoons at a time until you reach a smooth, silky sauce (about 2–3 tablespoons, more if needed).
Taste and adjust seasoning with additional salt and pepper if desired.
Transfer to a serving platter or plates, garnish with chopped fresh parsley and extra grated Parmigiano-Reggiano if you like, and serve immediately.