Bring a large pot of salted water to boil for the pasta. Meanwhile, heat the oil in a large skillet over high heat until just smoking. Add the broccoli and season with salt, to taste. Cover and cook until just crispy and brown around the edges, about 3 minutes. Remove the cover, stir, and reduce the heat to medium-high. Cover and cook until the florets are crisp but tender, about 5 minutes. Add the chicken stock and pepper flakes.
Add the capellini to the boiling water and cook until al dente, 2 to 3 minutes. Drain the pasta and add to the skillet with the broccoli and stock. Stir in half the cheese and season with salt and pepper. Divide among 4 warmed soup plates and sprinkle with the remaining cheese.