Add white wine, cherry brandy, garlic clove, cayenne, nutmeg, and cornstarch to a saucepan. Whisk to combine and place on the stove over medium heat. Cook, whisking occasionally, until the mixture thickens slightly and you can see wisps of steam in the pan.

Add Gruyère and Emmentaler cheeses, one handful at a time, slowly whisking after each addition until melted. Continue whisking until all cheese is melted and sauce is very smooth.