Preheat oven to 350 degrees F. Place beets in the center of 18-inch square of heavy foil. Drizzle with 1 tablespoon of the oil. 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 for 75 minutes or until tender. Cool for 30 minutes. Carefully open packet. Remove beets; gently transfer liquid to a measuring cup, adding enough water to equal 1/2 cup. Pour liquid into a medium saucepan. Cut beets into wedges.
In a 3-quart saucepan cook onion in remaining 2 tablespoons oil over medium heat until tender; add rice. Cook and stir for 5 minutes. Stir in dried basil, if using.
Meanwhile, add broth to beet liquid in saucepan. Bring to boiling; reduce heat to simmering. Carefully stir 1 cup of broth mixture into rice mixture. Cook, stirring frequently, over medium heat until liquid is absorbed. Continue to add 1/2 cup broth at a time, stirring frequently until broth is absorbed before adding additional broth (about 22 minutes).