Step 1

Line a large bowl with the butter slices, placing them in a single layer along the bottom and up the sides of the bowl; let stand at room temperature until the butter is softened.

Step 2

Meanwhile, in a food processor, process the cheese until very finely ground, about 40 seconds; transfer to a medium bowl (you should have about 1½ cups).

Step 3

In a large pot, bring 2 quarts water to a boil. Add the pasta and 1½ teaspoons salt, then cook, stirring often, until the pasta is al dente, about 2 minutes. Remove the pot from the heat. Using tongs, transfer the pasta from the pot, with ample water clinging to it, to the butter-lined bowl. Using the tongs, quickly stir and toss the pasta, incorporating the butter, until the butter is fully melted. Add ½ cup pasta water and toss until the water has been absorbed.

Step 4

Add 1 cup of the cheese, ⅓ cup at a time, tossing and adding the next addition only after the previous one has been incorporated. Next, toss in ½ to 1 cup more pasta water, adding about ¼ cup at a time, until the sauce clings to the pasta and only a small amount pools at the bottom of the bowl.

Step 5

Let stand for 2 minutes to allow the sauce to thicken slightly. If needed, toss in additional pasta water a little at a time until the sauce once again clings to the pasta and only a small amount pools at the bottom of the bowl. Taste and season with salt. Divide among warmed serving bowls and serve immediately with the remaining cheese on the side for sprinkling at the table.