Heat the butter in a 4-quart saucepan over medium heat. Add the onions and cook until they're tender-crisp. Add the potato and garlic and cook for 1 minute.
Drain the liquid from the beets and reserve 1 cup beet juice. Add the beets and broth to the saucepan and heat to a boil. Reduce the heat to low. Cover and cook for 15 minutes more or until the potato is tender.
Place 1/2 of the broth mixture into an electric blender or food processor container. Cover and blend until smooth. Pour into a medium bowl. Repeat the blending process with the remaining broth mixture. Return all of the pureed mixture to the saucepan. Stir in the reserved beet juice, dill and black pepper. Cook over medium heat until the mixture is hot.