The Day Before: Using butcher's twine, tie tenderloin at 1-inch intervals using butchers’ knots. Season generously with salt and pepper. Transfer to a wire rack set in a foil-lined rimmed baking sheet and refrigerate uncovered at least overnight and up to 2 nights.
When Ready to Cook: Adjust oven rack to center position and preheat oven to 225°F. Place baking sheet with rack and tenderloin in oven and roast until internal temperature registers 120 to 125°F on an instant read thermometer, 2 to 3 hours. Remove from oven and set aside at room temperature for 10 minutes. Cut and remove twine.
To Finish Under the Broiler: Adjust rack to 6 inches from broiler element and preheat broiler to high. Heat butter, swirling, in a medium skillet over high heat until foaming subsides and butter turns a light nutty brown. Add thyme and shallots and stir until crackling stops. Pour butter mixture over tenderloin and spread with a spoon until all surfaces are coated. Remove shallots from top surface of meat. Place pan with tenderloin under broiler and broil, turning every 30 seconds, until meat is well browned on all sides and internal temperature registers 125°F for rare or 130°F for medium-rare, about 2 minutes total.
To Finish on the Stovetop: Heat butter, swirling, in a medium skillet over high heat until foaming subsides and butter turns a light nutty brown. Add tenderloin, shallots, and thyme and cook, turning occasionally and spooning hot butter and aromatics over roast, until meat is well browned on all sides and internal temperature registers 125°F for rare or 130°F (54°C) for medium-rare, about 1 1/2 minutes.Transfer tenderloin to a cutting board and allow to rest for 5 minutes. Slice into 1/2-inch slices, sprinkle with chives and coarse sea salt, and serve with Horseradish Cream Sauce.