You'll need a stockpot and a 12-inch (30 cm) skillet with deep sides. We'll be working with both at the same time.
Bring a large pot of water to a boil. Generously salt and add the pasta. Boil until the pasta can fully bend while still undercooked, about 5 to 6 minutes. Reserve 2 cups (480 ml) of the pasta water before draining. Drain well.
Meanwhile, heat the olive oil in the skillet over low-medium heat. When hot and shimmery, add the garlic, stirring often until golden, about 3 minutes, taking care not to burn the garlic or it will taste bitter. Remove the skillet from the heat to carefully add the bottle of white wine, which may splatter at first. Add the broth and raise the heat to medium-high to bring the contents to a boil.
Add the parboiled pasta to the skillet and 1/2 cup (120 ml) of the reserved pasta water. Season with salt and freshly ground pepper. Keep the mixture at a boil and use tongs to toss frequently to prevent sticking.
Cook until the pasta is al dente, tender yet firm to the bite, and the liquid has been largely absorbed, though not entirely, tossing frequently, about 10 minutes. Add a little more of the reserved pasta water if the pasta absorbs too much. Keep a watchful eye as cooking times will differ depending on the type and brand of pasta.
While continuously tossing, add the grated cheese, a little at a time. By adding small amounts and tossing, the cheese will melt more evenly and will not stick to the bottom of the pan. Again, use more of the reserved pasta water, if necessary. Toss well. The pasta should be loose and silky and will continue to thicken upon standing.
Turn off the heat and add the butter, gently tossing to create a silky glaze. Taste and adjust the seasonings, if necessary.
To serve, drizzle lightly with olive oil, a few turns of freshly ground pepper and more grated cheese. Serve immediately.