Season the steak with salt and pepper. In a small bowl, combine the horseradish and mustard. Rub the mixture over the steak until well coated.
Heat a medium saute pan over medium high heat. Spray lightly with nonstick spray (I normally cook with a thin coating of real oil or butter, but the horseradish mixture will dissolve into the fat if there is too much present in the pan).
Saute the steak until it is well browned on both sides, and finished to your desired doneness (6-8 minutes for medium). Let rest for 10 minutes.
Meanwhile, heat the oil in a large pot until shimmering and hot. Add the shallots (if they bubble and begin to cook immediately, the oil is ready – if they burn, it is too hot). Fry until the shallots are golden and crispy, about 6-8 minutes. Dry on absorbant paper towels, and season well with salt.
Slice the steak. Spread additional creamy horseradish on the buns, if desired, and layer the steak, arugula, and crunchy shallots. Serve warm.
I always carry three kinds of hot sauce in my purse. I have a soft spot for making people their favorite dessert, especially if it's pie. My first cookbook, The Fearless Baker, is out on October 24, 2017.