Preheat oven to 375 degrees F.

Season beef tenderloin steaks with salt and pepper.

Heat a large nonstick skillet with oven safe handle over high heat.

Sear steaks in the hot skillet for 3 minutes on each side to caramelize.

Transfer the skillet to the preheated oven.

Cook for 8 to 15 minutes, depending on desired doneness (medium-rare to well-done).

While the steaks are cooking, melt 2 tablespoons of butter in a medium saucepan over medium heat.

Add flour to the melted butter and cook for 1 minute.

Whisk in milk until smooth.

Cook until the sauce thickens to gravy consistency.

Stir in prepared horseradish, chopped chives, salt, and pepper.

Transfer the horseradish sauce to a serving dish.

Remove the steaks from the oven and transfer them to a platter.

Return the skillet to the stovetop over moderate heat.

Deglaze the pan with 1/2 cup Madeira or dry sherry, scraping up any browned bits.

Add the remaining 1 tablespoon of butter to the pan.

Pour the pan drippings over the steaks to serve.