Melt the butter in a heavy saucepan over medium heat. Add the flour and whisk until thick and golden, 1 to 2 minutes. Stir in the milk and continue whisking until the mixture is thick and creamy, 2 to 3 minutes. Season with salt and pepper. Whisk the mixture into the chili until it is completely incorporated and there are no lumps.
Continue cooking the chili over low heat, stirring occasionally, for another 45 minutes to an hour. Generously season with salt, pepper and cayenne to taste. The chili is good straight from the pot, but is even better served 1 or 2 days after cooking.
Serve over cooked white rice with sour cream, freshly grated Cheddar and hot sauce on the side.