In a separate saucepan, melt the butter over medium heat. Sprinkle the flour into the pan and whisk immediately to evenly mix it into the butter. Cook for a minute or 2, and then pour in the milk and half-and-half. Continue cooking, whisking constantly, while the sauce thickens, about 2 minutes. Add salt, black pepper, cayenne, cardamom and and then add the grated cheese. Stir while the cheese melts. Turn off the heat.