Cook the sausage in a large pot over medium-high heat, breaking up the meat with the back of a spoon, until browned, 5 to 8 minutes. Using a slotted spoon, remove the sausage, leaving the fat behind.
Add 3 tablespoons oil and the bread crumbs to the pot; season with salt and pepper. Cook, stirring frequently, until the bread crumbs are golden brown, 2 to 3 minutes. Transfer to a bowl.
Cook the pasta in a large pot of salted boiling water until just al dente. Drain, reserving at least 1 cup of the pasta water.
Meanwhile, heat the remaining 3 tablespoons olive oil in the pot, then add the garlic and cook, stirring occasionally, until it’s toasted and golden brown, 2 to 3 minutes. Add the tomato paste and cook, stirring constantly, until it’s a nice brick-red color and starts to stick a bit to the bottom of the pot, 2 to 3 minutes more.
Reduce the heat to medium-low and add the broccoli rabe, then season with salt and pepper. Cook, stirring occasionally, until it’s all wilted and bright green, 3 to 5 minutes.
Add the pasta and chorizo back to the pot along with 1/2 cup of pasta water. Cook, stirring constantly, until each piece of pasta is coated nicely in the tomato-y, chorizo-y sauce.
Serve straight from the pot (or not), with toasted bread crumbs and pecorino for sprinkling over.
Alison Roman is a cook, writer and author of the bestselling cookbook "Dining In," published by Clarkson Potter in Fall 2017. Her second cookbook "Nothing Fancy," is now available. She is a bi-weekly columnist for the New York Times Cooking section, as well as a monthly contributor to Bon Appétit Magazine. Originally from Los Angeles, she lives in Brooklyn until she decides to move upstate like everyone else.