Season meat with salt and pepper. Heat a large nonstick skillet with oven safe handle over high heat. If you have a rubber handled pan, wrap the handle twice with tin foil for baking.
Sear meat in the very hot nonstick skillet to caramelize the beef, 3 minutes on each side. Transfer to oven and cook 8 to 15 minutes longer for medium rare to well done meat.
In a medium saucepan over medium heat melt 2 tablespoons butter. Combine melted butter with flour and cook a minute longer. Whisk in milk. Thicken milk to gravy consistency, then stir in horseradish, chives and salt and pepper. Transfer sauce to a serving dish.
Remove meat from oven and transfer to a platter. Return pan to stovetop over moderate heat and deglaze pan with 1/2 cup Madeira or dry sherry. Add remaining tablespoon butter to the pan and pour the drippings over steaks.