Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the sausage and cook until browned, breaking up with a wooden spoon, about 8 minutes.
Add the broccoli, red pepper flakes, salt, and pepper and stir to combine.
Add the pasta and chicken stock and bring to a boil. Cook for 10–12 minutes, stirring constantly, until the liquid is absorbed and the pasta is tender. Add the Parmesan and stir to incorporate.
Divide the pasta between serving bowls and garnish with more Parmesan.