Combine the oil, vinegar, apple juice, thyme, pepper, salt and garlic in a small bowl and whisk to blend. Season with additional salt and pepper.
Preheat the oven to 425 degrees F. Place the goat cheese in a medium bowl. Drizzle with just enough dressing to moisten. Refrigerate until ready to serve.
Wrap each beet in foil. Place the beets directly onto the oven rack and roast until tender, about 50 minutes. Eight minutes before the beets are tender, place the green beans in a pie dish. Add the oil and a pinch each salt and pepper and toss to coat. Roast the green beans alongside the beets until crisp-tender, about 8 minutes. Remove the foil from the beets and set aside until cool enough to handle. Cut each beet into 6 to 8 wedges.
Combine the lettuces in a large bowl. Add the green beans and apple slices and top with the beet wedges. Sprinkle the goat cheese on top. Just before serving, pour the dressing over the salad and toss until coated.