Make the compound butter: Place the capers, shallot, chives, tarragon, ½ cup butter, and pepper in the bowl of a small food processor and process until well blended.
Scrape the butter out on the center of a piece of plastic wrap, and use the plastic wrap to form the butter into a 4-inch long log about 2 inches in diameter. Wrap it well, and refrigerate for at least 2 hours and up to 5 days.
To cook the tenderloin, preheat the oven to 475 F. Pat the beef dry with paper towels, and place it on a rimmed baking sheet. Rub the two tablespoons butter all over the meat, and evenly sprinkle the salt and pepper over the surface.
Place the beef in the oven and roast for about 23 to 25 minutes for rare (an internal temp of 120 F), and 27 to 30 minutes for medium-rare (an internal temp of 125 F), depending on the thickness of the meat. Let the meat sit on a cutting board, tented with foil, for 20 minutes before slicing it into thick slices (or thin, your call!). Remove the compound butter from the fridge while the meat rests.
Place a slice of the compound butter on top of the warm slices of meat as you serve them.
Notes
Compound butter will last in the fridge for a week. You can also freeze compound butter for up to a month.