Roast the beets: Preheat the oven to 325 degrees F. Place the beets in a baking dish, then add 1 cup water. Cover tightly with foil. Roast until the beets yield when pierced with the tip of a knife, 1 to 1 1/2 hours, depending on the size of the beets. Set aside to cool.
Make the vinaigrette: In a medium bowl, whisk together the mustard, lemon juice and red wine vinegar. Slowly whisk in the olive oil. Season with kosher salt. Set aside.
Make the walnuts: Heat a medium skillet over medium heat and add the olive oil. When it gets hot (but not too hot), add the walnuts and season with kosher salt. They should fry slightly in the oil. When you can smell the nuts, add the paprika and reduce the heat to low. Cook for 2 to 3 minutes, stirring constantly, until the nuts brown slightly. Drain on paper towels and set aside.
Assemble the carpaccio: Lightly brush a platter (or individual plates) with olive oil. Rub the skin off of the beets with paper towels; trim any root tips and cut the beets into thin slices. Place in a single layer on the platter (or plates). Season with the sugar and flaky salt. Drizzle with the vinaigrette and top with the walnuts. Snip the chives into 1-inch pieces over the beets.