Position a rack in the center of the oven and preheat to 475 degrees . Rub the beef with the EVOO and a generous amount of salt and pepper. Lay the onion slices in a single layer in the roasting pan, then top with the beef, fat side up. Roast for 15 minutes. Reduce the heat to 375 degrees and roast for 20 minutes more. Add the carrots and parsnips and baste the meat with the pan juices. Cook the meat until it registers 125 degrees on a meat thermometer, about 1 hour 10 minutes for medium-rare, basting again halfway through.
Meanwhile, in a large bowl, form a well in the center of the flour. Add the eggs to the well and stir into the flour using a wooden spoon, gradually pulling them together. Slowly pour in the milk and beat well for 5 minutes. Add 3 tbsp. chives and 1 tsp. salt; cover and set aside for at least 30 minutes.
In a small bowl, mix together the sour cream, horseradish and remaining 1 tbsp. chives; season with salt. Refrigerate.
Remove the roast from the oven and transfer the meat and veggies to a platter; tent with foil. Pour the pan drippings into a small bowl and set aside; reserve the roasting pan. Let the beef rest for at least 15 minutes before carving.
Increase the oven heat to 425 degrees . Spoon 1 tsp. of the roast drippings into each cup of a 12-cup muffin pan. Put the muffin pan in the oven and let heat until very hot, about 5 minutes. Give the chive pudding batter a good stir. Carefully remove the muffin pan from the oven and pour the batter evenly among the cups (about 3 tbsp. batter per cup). Bake until puffed, golden-brown and crisp, 20 to 25 minutes. Do not open the oven for the first 15 minutes of baking (if you do, the puddings wont rise). While the puddings bake, set the roasting pan over medium heat, add the cooked vegetables and rewarm for a few minutes.
Carve the roast and serve with the vegetables, puddings and chilled horseradish sauce.