To make the polenta fries, first boil the water with the salt. Gradually add the corn flour, stirring constantly with a whisk. As soon as the mixture thickens, use a rubber spatula or wooden spoon.
Reduce the heat and simmer gently. Stir frequently to prevent lumping. When the mixture begins to pull away from the sides of the pan, it is ready. This will take about 10 to 15 minutes. Sometimes it won't release from the sides. In that case, you can assume it is ready after 15 minutes. Remove from the heat and stir in the cheese, rosemary and butter.
Quickly pour the mixture onto a greased baking sheet lined with parchment paper. Spread it out in a layer about 1 cm (0.5 inch) thick. Smooth the top with a spatula and let it cool. It will cool a little faster in the refrigerator.
Heat the oil in a frying pan to 150˚C (300˚F). Remove the corn flour mixture from the baking sheet and turn out onto a board. Cut the fries into 1 cm (0.5 inch) thick slices.
Sprinkle the fries with a little flour, shake off excess flour and fry in the oil until light brown, about 5 minutes. Drain on paper towels.
Increase the temperature of the oil to 175˚C (350˚F) and fry again for 2 minutes or until golden brown and crisp.
Drain again on paper towels.
Serve with spicy mayonnaise and some more grated parmesan.