Bring a large pot of salted water to a boil. Add the broccoli florets and stems and cook until tender but still bright green, about 3 minutes. Remove to a bowl using a slotted spoon; set aside. Return the water to a boil, then add the pasta and cook as the label directs. Reserve 2 cups cooking water, then drain the pasta and set aside.
Heat the olive oil in the empty pot over medium heat. Add the garlic and cook until slightly softened, about 1 minute. Add the broccoli stems, half of the florets and 1 cup of the reserved cooking water. Cover and cook until very soft, about 12 minutes.
Return the pasta to the pot along with another 1/2 cup of the reserved cooking water. Add the remaining broccoli florets, half of the ricotta salata, and the herbs and scallions; stir to combine, adding more cooking water to loosen, if needed. Top each serving with the capicola and the remaining ricotta salata.