Meanwhile, in a small pan, sauté the chopped shallot and thinly sliced potato in a drizzle of olive oil. Season with salt, then add a bit of the reserved cooking water. Cover and cook until everything is soft. Blend the broccoli florets with basil and a little cooking water using an immersion blender. Then add the potato and shallot mixture, Parmesan, and a steady stream of olive oil. Continue blending until smooth, adjusting the consistency with more cooking water as needed. Season with salt to taste.

Cut the monkfish meat into small cubes, keeping the central bone. In the pan you’ll use for the pasta, heat a drizzle of olive oil, add the deseeded chili pepper, the fish bone, garlic, and a sprig of basil. Let everything infuse for a bit, then remove the garlic, bone, and basil. Meanwhile, cook the pasta in the reserved broccoli cooking water.

About 2–3 minutes before the pasta is done, transfer the linguine to the pan. Let it absorb the flavors, then continue cooking, adding more cooking water as needed. When the pasta is al dente, add the broccoli cream, mixing well and adding a little more water if necessary (it tends to thicken). Then add the monkfish, olives, another drizzle of olive oil, and toss everything together.