Brush cookie sheet with vegetable oil and set aside.
In a medium saucepan, heat water to boil and add salt. Whisking vigorously, pour in polenta slowly in a thin stream and then cook until quite thick, about 1 minute, switching to a wooden stick to stir as the polenta thickens. Pour polenta out on to cookie sheet and spread out flat until it covers the sheet, about 1-8- to 1/4-inch thick. Allow to cool 1 hour.
Rinse anchovies and remove heads, fins and bones. In a small saucepan, heat olive oil over medium heat with garlic and cook until just light brown. Add anchovies and capers and stir until broken up into 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 oil is heating, cut polenta into 2-inch rounds using a grappa glass or cookie cutters, yielding about 40 pieces. Place 1/4 teaspoon anchovy mixture on one polenta disk and make a sandwich by covering 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 and serve hot.