Preheat the oven to 400˚. Scrub the beets and put them on a large piece of foil; season with salt and pepper, add 3 thyme sprigs and drizzle with 3 tablespoons olive oil. Bake until the beets are tender, about 1 hour. When the beets are cool enough to handle but still warm, slip off their skins and chop them into large chunks.
Bring the chicken stock to a gentle simmer in a saucepan over medium-low heat. Meanwhile, heat the remaining 3 tablespoons olive oil in a large heavy-bottomed pot over medium heat. Add the onion, carrots, garlic and remaining 3 thyme sprigs and cook, stirring, until softened and just starting to color, about 10 minutes.
Add the warm chicken stock to the vegetables in the pot and simmer until the vegetables are tender, about 20 minutes. Remove the thyme sprigs. Put the chopped beets in a blender and add the cooked vegetables and stock. Blend until smooth, adding more stock if the puree is too thick. Add the vinegar and honey; season with salt and pepper. Blend again to incorporate the flavors. (The borscht can be served hot or cold; for a chilled soup, transfer to a bowl and refrigerate until cold.)
To make the garnish, grate the apple on the large holes of a box grater into a medium bowl; mix in the dill. Ladle the soup into bowls and swirl in a big dollop of sour cream. Top with the apple-dill mixture.