Place potatoes in a large pot of cold water and bring to a boil. Salt the water, lower the heat and simmer until fork-tender, about 15 minutes. Drain, then mash with the milk until smooth. Stir in the egg and the horseradish and season with salt and pepper.
In a large, deep skillet, heat the extra-virgin olive oil, 2 turns of the pan, over medium heat. Add the parsnips, carrots, onion and bay leaf. Season with salt and pepper and cook, stirring, until crisp-tender, 8 to 10 minutes.
Crumble the beef into the pan and cook, stirring often, for 5 minutes. Sprinkle the flour on top and cook, stirring, for 1 minute. Stir in the beef broth and cook until thickened, 1 to 2 minutes. Season with Worcestershire sauce and plenty of salt and pepper. Discard the bay leaf.
Preheat the oven to 450 degrees. Grease a shallow medium casserole dish, add the meat mixture and top with the horseradish potatoes. Bake until golden, 5-10 minutes. Sprinkle the chives on top.