Before cooking, remove the roast from the refrigerator and allow to sit for 30 minutes. (Eye of round roast is a dense piece of meat, so taking the chill off will help it cook more evenly.)
Heat the oven to 375F. Heat the oil in a large cast iron pan over high heat. Pat the roast dry with paper towels and season all over with the salt and black pepper.
When the pan is hot, add the roast and sear undisturbed for 2-3 minutes on both sides. Transfer the pan to the oven and continue to cook for 30-40 mins, flipping the roast once halfway through cooking. The internal temperature of the roast should read 125 for rare. Allow the meat to rest 10 minutes on a cutting board, then slice the roast thinly across the grain.
While the beef roasts, assemble the horseradish cream. Drain the horseradish through a fine mesh sieve, pressing out as much liquid as possible. Combine with the remaining cream ingredients in a medium bowl.
To assemble the crostini, layer 1 piece of beef atop the toast, followed by a teaspoon or so of the cream. Place one chive piece on top and sprinkle with sliced chives. Serve right away and enjoy!