Toss all three cheeses and 1 tablespoon cornstarch together in a large bowl. In a separate bowl, whisk the rest of the cornstarch with the brandy.
Pour wine into a saucepan and set over medium heat. When wine is just about to boil, bubbles will appear on the edges of the liquid; turn the heat down to medium-low and add 1 cup cheese. Stir to melt, then repeat with remaining cheese.
Pour brandy into the melted cheese and wine mixture and season with nutmeg, paprika, and cayenne.
Light the burner under a fondue pot and pour the fondue into the pot. Serve fondue with long forks and bread, cooked potatoes, and vegetables for dipping.