Combine the beets, tomatoes, garlic, cloves, 4 cups water, the chicken broth, 1/2 teaspoon salt and a few grinds of pepper in a pot. Bring to a boil, then reduce the heat to medium low; simmer until the beets are tender, about 30 minutes.
Working in batches, puree the soup in a blender until smooth; transfer to a large bowl. Whisk in 1/2 cup yogurt and season with salt and pepper. Let the soup cool slightly, then cover and refrigerate until cold, about 2 hours.
Mix the remaining 3/4 cup yogurt with the horseradish in a small bowl; season with salt. Thin the soup with a little water, if necessary. Serve in shot glasses; top with the horseradish yogurt and dill.
Photograph by Christina Holmes
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.