1. Stir cheese into hot basic polenta. Add salt and pepper to taste. Scrape mixture into a buttered 8-inch square baking pan and, with a spatula that has been dipped in water, spread level, 3/4 to 1 inch thick. Chill until very firm, at least 1 1/2 hours and up to 1 day.
2. Run a knife around edges of pan to loosen polenta, invert onto a board, and lift off pan. Trim off any rounded edges from polenta (save trimmings for another use). Cut polenta in half to form two equal rectangles (wipe knife with a wet paper towel as needed to prevent sticking), then cut each half into strips 1/2 inch wide and 3 to 4 inches long to produce thick fries (or cut strips in half through wider side to produce thin, square-cut fries). Trim away uneven edges. Spread about 1/2 cup flour on a rimmed plate and roll each polenta strip lightly in flour.
3. Pour 3 inches oil into a deep-fryer set at 375° (or medium-high) or a deep, heavy, 3- to 4-quart pan over medium-high heat. When oil reaches 375°, working in batches, lower polenta strips into oil in a deep-fry basket or a few at a time with a slotted spoon; fry until crisp and golden, 7 to 9 minutes. Remove with basket or slotted spoon. Drain briefly on paper towels, then transfer, in a single layer, to baking pans and keep warm in a 200° oven.
4. Sprinkle fries with salt and serve with spicy tomato dipping sauce.