Step 1: In a small cup or bowl mix mayonnaise and horseradish together and set aside.
Step 2: In a large skillet over medium high heat melt 1 tablespoon of the butter.
Step 3: Sprinkle steaks with salt and pepper. Add steaks to skillet and cook to desired doneness (5-10 minutes per side). Using tongs transfer steaks to work surface.
Step 4: In the same skillet melt 3 tablespoons butter over medium-high heat and add onions; sauteing until dark brown (20-30 minutes). Add mushrooms and saute until tender (5 minutes). Stir in broth and boil until juices are reduced to a glaze. Season onion/mushroom mixture with salt and pepper.
Step 5: Place one croissant bottom, cut side up, on each of 4 plates. Spread bottom with horseradish mixture and top with 1/2 cup arugula. Thinly slice steaks and divide among croissants. Top each with onion mixture. Spread remaining horseradish mixture on cut side of crossants tops. Place tops on sandwiches.
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