Bring a large pot of generously salted water to a rolling boil. Cook the tagliatelle according to package directions until al dente.
While the pasta cooks, peel the garlic and grate it on a microplane into a small pile.
In a large, cold skillet, add the butter and the microplaned garlic. Turn the heat to medium-low. Let the butter melt and the garlic gently infuse for about 2 minutes. Do not let the garlic brown.
Pour in the heavy cream, stir, and let it come to a gentle simmer. Let it cook for 2–3 minutes, just until it thickens slightly. Taste and add a pinch of salt.
Before draining the pasta, scoop out 1 cup of the starchy pasta water. Set it aside.
Use tongs to transfer the drained pasta directly into the skillet with the cream sauce. Add a splash of the reserved pasta water and toss vigorously. The sauce will go from thin to thick and glossy.
Turn off the heat. Add the grated Parmesan and toss until it melts into the sauce. The residual heat is enough to melt it perfectly without turning it grainy.
Squeeze the half lemon over the pasta. Toss again. Taste and adjust salt.
Use tongs to create a neat nest of pasta in a wide, shallow bowl. Spoon a little extra sauce from the pan over the top.
Sprinkle with fresh parsley and a generous pinch of flaky salt. Serve immediately.