Preheat oven to 350 degrees F. Cut off beet stems and roots; if desired, leave about 1/4 inch of the stems on the beets. Chop and reserve 1 cup beet stems. Peel beets.
In a medium bowl, combine beets, chopped stems, olive oil, garlic, rosemary, thyme, salt, and pepper. Toss to coat evenly. Transfer beet mixture to the center of a 16-inch square piece of heavy foil. Bring up opposite edges of foil and make a double fold. Fold remaining edges together to enclose the beets, leaving space inside for steam to build.
Roast, folded side up, about 1 hour or until beets are tender. Cool slightly. Unfold foil packet. Divide beet mixture and goat cheese among four to six appetizer plates. Drizzle with balsamic vinegar. Makes 4 to 6 servings.