How to Make It
Place the water, salt and butter in a 2-quart saucepan and bring to a boil. Have a whisk and a wooden spoon ready. Remove the pan from the heat and dump the flour in all at once, whisking it together. Return the pan to the burner and start stirring with spoon. Cook for 3 minutes, stirring constantly, until the dough starts to pull easily from the sides of the pan and form a ball. Remove from the heat and stir until tepid, 6-8 minutes. Add the eggs, one at a time, stirring in each one completely before adding the next. Pour the resulting mixture into a mixing bowl and cover with plastic wrap, using your fingers to lay the plastic wrap so that it rests directly on the top of the batter. Set aside at room temperature.
Bring 4 quarts of water to boil and set up an ice bath next to the stove with 3 cups of ice and 4 cups of water. Plunge the cauliflower into the boiling water and cook until tender, not al dente, about 8-10 minutes. Remove the cauliflower and refresh in the ice bath. Drain the cauliflower pieces well, dry on a kitchen towel and set aside in a large bowl. Remove the plastic from the batter, finely chop the anchovy fillets and add to the batter. Stir well to distribute evenly.
Preheat the oven to 200°F. Toss the cauliflower into the batter and stir gently. In a 6- to 8-quart pot, heat the olive oil to 375°F. Set up a baking sheet in the oven with several layers of paper towels to drain the fritters and keep them warm. Using 2 tablespoons, form fritters by grabbing 1 floret of cauliflower at a time with a generous amount of the batter that adheres to it and carefully dropping them one a time into the hot oil. Cook 6 or 7 at a time until golden brown, flipping them with a slotted spoon and remove to drain on the paper towels, then place on the sheet pan in oven while the rest are cooking. Serve hot.