Season beef with salt and black pepper.

Heat oil in a large skillet over medium-high heat. Cook beef roast in hot oil until browned on all sides on each side, 4 to 5 minutes per side. An instant-read thermometer inserted into the center should read 125 degrees F (52 degrees C). Transfer beef to a plate to rest.

Remove skillet from heat and whisk flour into skillet until flour taste cooks off, about 1 minute. Pour cold beef broth into skillet; place skillet over medium-high heat. Stir balsamic vinegar, black pepper, and any accumulated juices from the roast beef plate into beef broth mixture; cook until jus reduces slightly, about 5 minutes. Season with salt.

Slice sirloin as thinly as possible against the grain. Pile meat onto the base of each roll. Spread a thick layer of horseradish on the top bun and place top bun over meat. Serve with jus for dipping.