Line a 9-inch x 12-inch rimmed, nonstick baking sheet with parchment paper.
Measure 2/3 cup polenta and reserve in shallow dish.
Bring the stock to a low boil in a saucepot. Reduce the heat to a simmer and whisk in the remaining 1 1/3 cups quick-cooking polenta; whisk to thicken, 2-3 minutes. Melt in the butter, stir in 1/2 cup cheese, then season with salt, pepper and nutmeg, to taste.
Pour the polenta onto the baking sheet and smooth with a wet frosting spatula or rubber spatula – the layer should be no more than three-quarters of an inch thick. Let cool for 1 hour.
Invert the pan and remove the paper from the polenta. Cut 3-inch fries about half-inch thick.
Heat a few inches of oil in a Dutch oven or tabletop fryer to 350°F.
Add the fine breadcrumbs or flour and 1/2 cup cheese to the remaining polenta reserved in dish. Toss a quarter of the fries at a time in the coating and fry until golden, 3-4 minutes. Let cool on a wire rack; hold warm on the wire rack over a baking sheet in a 275°F oven until ready to serve.