For the citrus simple syrup: Preheat the oven to 350 degrees F.
Add the sugar, lemon juices and lime juice to a small saucepan over medium heat. Bring to a low simmer and stir to dissolve the sugar, about 5 minutes. Turn off the heat and set aside.
For the beets: Place a layer of rock or kosher salt in a baking pan. Nestle the beets on top of the salt and bake until soft, about 1 hour. Let cool. Cut off skin and roughly chop.
For the assembly: Place the beets in a blender along with half of the vodka and 2 tablespoons simple syrup. Puree until smooth. Add the remaining vodka and ice cubes and puree again. Serve in coupe glasses.