Place beets in a medium saucepan with water to cover. Bring to a boil over high. Reduce heat to medium, and simmer, uncovered, until fork-tender, about 20 minutes. Drain; cool 10 minutes. Peel beets, and cut into quarters (or cut into eighths if using larger beets). Place in a large mixing bowl.

While beets are cooking and cooling, place potatoes, garlic, thyme, and 1 tablespoon of the salt in a medium saucepan. Add water to cover, and bring to a boil over high. Reduce heat to medium, and simmer, uncovered, until potatoes are fork-tender, about 10 minutes. Drain, discarding garlic and thyme; cool about 20 minutes. Add potatoes to cooled beets in bowl; refrigerate until thoroughly chilled, about 30 minutes.

Whisk together yogurt, sour cream, lemon zest and juice, vinegar, pepper, 1 tablespoon of the dill, and remaining 1 teaspoon salt in a small bowl until smooth and combined. Add to chilled potato-beet mixture, and toss to coat. Transfer to a serving bowl. Sprinkle with remaining 1⁄2 tablespoon dill, and serve immediately.