1. Preheat oven to 400°.
2. Rinse beets thoroughly. Leave 1-inch stem on beets; reserve greens for another use. Arrange the beets, 1/4 cup water, garlic, and thyme in a 13 x 9-inch glass or ceramic baking dish. Cover with foil; bake at 400° for 1 hour or until tender. Uncover dish; cool completely. Reserve garlic and pan juices; discard thyme. Peel beets, keeping stems intact. Cut beets lengthwise into quarters.
3. Squeeze garlic pulp into a mini chopper; discard peels. Add buttermilk and next 6 ingredients (through goat cheese); process until smooth. Spoon 3 tablespoons sauce onto each of 6 plates; arrange 8 beet quarters over sauce. Drizzle 1/2 teaspoon pan juices over each serving; garnish with dill sprigs, if desired.