Open-Faced Burgers with Horseradish-Cheese Sauce

In a small saucepan combine the garlic paste and the butter and heat the mixture over moderately low heat, stirring, until the butter is melted. Brush the cut sides of the roll with the butter mixture and toast the roll halves lightly. In a saucepan combine the sour cream and the Cheddar, heat the mixture over moderately low heat, stirring occasionally, until the cheese is melted, but do not let it boil, and stir in the horseradish, the pickle, and the jalapeño chili. In a heavy skillet (preferably cast-iron) heat the oil over moderately high heat until it is hot but not smoking and in it cook the patties, seasoned with salt and black pepper, turning them once, for 8 minutes for medium-rare meat. Transfer the burgers to the roll halves and spoon the horseradish-cheese sauce over them.