Brush a large cookie sheet with olive oil and set aside.
In a medium saucepan, heat the water to a boil and add the salt. Whisking vigorously, pour in the polenta, slowly, in a thin stream, and cook until quite thick, about 1 minute, switching to a wooden spoon to stir as the polenta thickens. Pour the polenta out onto a cookie sheet and spread out flat until it covers the sheet, about 1/8-inch to 1/4-inch thick. Allow to cool 1 hour.
Rinse the anchovies and remove the heads, fins and bones. In a small saucepan, heat 1/4-cup olive oil over medium heat with the garlic and cook until just light brown. Add the anchovies and capers and stir until broken up into a paste. Remove from heat and set aside.
Pour 3 inches of oil into a 6-inch tall frying pan and heat to 360 degrees F. While the oil is heating, cut the polenta into 2-inch rounds, using a grappa glass or cookie cutter, yielding about 40 pieces. Place 1/4-teaspoon anchovy mixture on 1 polenta disk and make a sandwich by covering it with a second polenta disk. Dredge each sandwich in flour, then in egg and fry until golden brown, about 3 minutes. Continue until all the sandwiches are finished.
In a medium bowl, combine the puntarelle, remaining 1/4-cup olive oil, lemon juice and the remainder of the anchovy mixture. Season with salt and pepper, to taste, and toss well to coat. Serve with the little sandwiches.
Recipe copyright 2000, Mario Batali. All Rights Reserved.