Place the water, salt, and butter into a 2-quart saucepan and bring to a boil. Have a whisk and a wooden spoon ready. Remove the pan from the heat and add the flour all at once, whisking it in. Return the pan to the burner and start stirring with a spoon. Cook 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 to 8 minutes.

Add the eggs, 1 at a time, stirring each one completely in before adding the next. Pour into a mixing bowl, cover to the surface with plastic wrap and 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 water and cook until tender, not al dente. Remove the cauliflower and refresh in the ice bath. Drain cauliflower pieces well, dry on a kitchen towel and place in a bowl.

Finely chop the anchovy fillets and add to the cauliflower pieces. Stir well to distribute.

Preheat the oven to 200 degrees F.

Remove the plastic from the batter, pour the cauliflower mixture into the batter, and stir gently to mix well.

In a 6 to 8-quart pot, heat the olive oil to 375 degrees F. Set up a tray 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, swirling it so a generous amount of the batter that adheres to it, and dropping them 1 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. Serve hot.