Trim beets, leaving roots and 1-inch stems. Drizzle beets and shallot evenly with 2 tablespoons oil, and wrap individually in foil.
Bake at 425° for 1 hour or until tender; cool. Trim stems and roots, and rub skins to remove. Finely chop beets, shallot, and thyme, and place in a medium bowl. Stir in remaining 2 tablespoons oil, salt, and pepper.