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.