Bring a large pot of water to a boil. Trim the cauliflower into bite-size florets. Salt the water and cook cauliflower about 3 minutes. Using a slotted spoon or spider skimmer, transfer to a plate to cool. Return the water to a boil.
In a large skillet, heat the EVOO, four slow turns of the pan, over medium. Add the tuna, shallots, garlic, parsley stems and chile. Cook, stirring, 2 to 3 minutes, then add the vermouth and reduce the heat to low.
Cook the pasta to al dente, adding about 1/2 cup pasta cooking water to the tuna mixture. Drain the pasta and add to the tuna mixture along with the parsley leaves and cauliflower. Toss and season with sea salt to taste.