For the burgers: In a medium bowl, mix together the tomato paste, horseradish, celery seeds, hot pepper sauce, Worcestershire sauce, salt, and a generous grinding of black pepper. Add the beef and mix thoroughly with your hands, making sure the seasonings are spread throughout the meat. Let stand for at least an hour for the flavors to develop. Meanwhile, make the horseradish dressing.
For the dressing: Mix the grated horseradish, creme fraiche and vinegar together in a bowl, seasoning with salt and pepper to taste. Break off a small piece of the beef mixture, the size of a walnut, and fry in a little oil until well cooked. Taste and adjust the seasonings, if necessary.
To serve: Heat a grill. Shape the meat into 4 patties about 3/4-inch thick. Lightly brush each one with oil, then place on a grill over high heat. It's hard to give exact cooking instructions due to variables such as the thickness of the burger, the heat of the grill, and personal preference. For a medium-rare burger, you will probably need to grill for 3 to 4 minutes, then flip over and cook the other side for a couple of minutes. If you're cooking the burgers indoors, use a lightly oiled grill pan set over medium-high heat. Let the burgers rest for a minute or two while you prepare the buns.
Toast the buns or other bread lightly on the cut side, then top with the lettuce leaves and tomato slices. Add the burgers and smear with horseradish sauce, and a dollop of ketchup if you like. Top with the other half of the bread.
Good with French fries, obviously, but if they're too much effort, really good potato chips will do.
Reprinted with permission from River Cottage Every Day by Hugh Fearnley-Whittingstall, copyright (C) 2009. Published by Ten Speed Press, a division of Random House, Inc.