Preheat the oven to 375 degrees F. Sprinkle the filet generously all over with kosher salt and pepper. Melt the butter in a medium-sized ovenproof skillet over medium heat until the foam subsides. Cook the filet until well browned all over, about 3 minutes per side. Transfer the pan to the oven and bake until an instant-read thermometer inserted into the center registers about 130 degrees F, about 10 minutes. Transfer to a cutting board and let rest until room temperature, at least 15 minutes. Thinly slice across the grain.
Stir together the horseradish, mayonnaise and sour cream. Spread the bread slices generously with the horseradish mixture and place a slice of beef decoratively on top. Sprinkle with a little sea salt, and then top with 2 crossed chives.
The beef can be cooked a day ahead. Slice just before assembling.