Melt butter in a large pot over medium heat. Stir in garlic and flour; cook and stir until garlic is fragrant but not brown, about 1 minute.

Whisk in heavy cream and milk until mixture is hot and slightly thickened, about 10 minutes. Gradually stir in Parmesan cheese and fontina cheese. Season with salt and pepper.

Continue to simmer until cheese melts and sauce thickens, stirring often, 20 to 30 more minutes.