Make the horseradish cream: Mix the horseradish, mayonnaise, sour cream, mustard, sugar, 3/4 teaspoon salt and 1/4 teaspoon pepper in a bowl. Cover and chill at least 2 hours.
Make the sliders: Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the bun bottoms, cut-side down, and cook until golden; set aside. Melt 1 more tablespoon butter in the skillet and toast the bun tops until golden; set aside.
Season the beef generously with salt and pepper. Heat the remaining 2 tablespoons butter in a large cast-iron skillet over medium-high heat. Working in batches, add the beef and cook until browned but still pink in the center, about 2 minutes per side. (You can also cook the tenderloin in one piece: Brown on all sides, then roast in a 425 degrees F oven, 10 to 15 minutes. Let rest, then slice into 12 pieces.)
Assemble the sliders: Put a slice of beef on each bun bottom. Top each with a generous spoonful of the horseradish cream, some onion rings and a bun top.