Bring a medium pot of well-salted water to a boil. Cook the pasta until just barely al dente, about 1 minute less than package timing. Reserve 1 cup of the cooking water, and drain the pasta.

While the pasta cooks, put the pancetta in a cold 10-inch skillet and set over medium-high heat. When the pancetta starts sizzling, add the garlic and cook, stirring constantly, until starting to brown, 1 minute. Reduce the heat to medium and continue to cook the pancetta until golden but still chewy at the center (taste a piece if you're not sure), an additional 2 to 3 minutes. If the pancetta has rendered a lot of its fat, spoon off all but 1 tablespoon of the fat from the pan.

Add the beans to the pan and cook, stirring constantly, until they're crisp-tender, 3 to 4 minutes. Remove the garlic and season the beans with salt and pepper. With the pan still over medium heat, add the pasta, 1/2 cup of the pasta water, and the butter. Toss to combine. Add another 1/4 cup pasta water and 3/4 cup of the Parmigiano. Stir well and season to taste with salt and pepper. If necessary, add a little more pasta water to loosen the sauce.

Transfer the pasta to a serving bowl. Grind black pepper over the top and sprinkle with the remaining cheese.