Cut the greens off the beets and set in water so they don’t wilt. Boil the beets in salted water to cover until you can insert a fork into the center of the largest, about 20 minutes. Drain the beets and let them sit until cool enough to handle.
Peel the beets (their skins should slip off easily after being boiled) and chop them coarsely. Remove the greens from the water and dice them.
In a large pot over medium heat add oil and the onions and cook, stirring occasionally, until the onions are just turning soft, about 5 minutes.
Add the garlic, and cook until fragrant, about 1 minute.
Stir in the cabbage and cook, stirring occasionally, until the cabbage wilts, 3 to 5 minutes. Add the beet greens and stir until they are just wilted, 1-2 minutes.
Add the broth and bring to a boil; reduce the heat to maintain a steady simmer, add the caraway seeds and chopped beets, and cook until the vegetables are tender and flavors blend, 15 to 20 minutes.
Add salt and lemon juice, if using, to taste. Serve hot, with a dollop of yogurt or sour cream and a sprinkle of fresh herbs.