Heat oven to 400 degrees F.
Scrub the beets and put them on a large piece of aluminum 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. Set aside. When the beets are cool enough to handle but still warm, slip off their skins, and chop them into large chunks.
In a large heavy bottomed pot over medium heat, add the remaining 3 tablespoons olive oil. Put in the onion, carrots, garlic, and remaining 3 thyme sprigs and cook until softened and just starting to color, about 10 minutes. Add the chicken stock and simmer until the vegetables are tender, about 20 minutes. Remove the thyme sprigs. Put the chopped beets into a blender and add the cooked vegetables and most of the stock. Blend until smooth, add more stock if the puree is too thick. Add the vinegar and honey; season with salt and pepper. Blend again to incorporate flavors. Borscht can be served hot or cold.
To make the garnish, grate the apple on the large holes of a box grater and mix in the dill. Serve in bowls, garnished with a big dollop of sour cream and topped with the apple and dill mixture.