In a small bowl, whisk together the black pepper, dry mustard, kosher salt, and cayenne pepper. Pat dry the beef tenderloin with paper towels. Tuck the tail of the tenderloin under the roast, to create a more uniform roast. Tie the roast with kitchen twine. Rub the roast with the olive or vegetable oil. Sprinkle seasoning mix over the whole roast. Allow the roast to rest at room temperature for 1 hour.
Turn on the burners of your gas grill to high heat. Preheat the grill with the lid down until it is very hot, about 15 minutes. Scrape the grill grate clean with brush. Oil the grate with oil-dipped paper towels and tongs to reduce sticking.
Leave the primary burner on and turn off other burners. Place the tenderloin on the hot side of the grill over the primary burners. Cover and grill until the roast is well browned on all sides, about 3-4 minutes per side.
Transfer the tenderloin to the cooler side of the grill and cook covered until the meat registers between 125-130 for medium rare, about 20-25 minutes. Transfer to a cutting board, tent with foil and allow to rest for 15 minutes before slicing.
In a bowl, whisk together the ingredients for the horseradish chive sauce. Serve along side the tenderloin.