Toss together carrots and onions. Spread evenly in half of a 13x9x2-inch baking pan. Place beets in opposite half of pan. Drizzle 1 tablespoon of the olive oil over vegetables. Cover with foil and bake in a 400 degree F oven for 40 to 45 minutes or until tender. Transfer carrots and onions with a slotted spoon to a storage container; cover. Transfer beets to a separate storage container; cover.
Add 1 tablespoon hot water to baking pan, stirring to dislodge any browned bits. For dressing, combine liquid in pan, the remaining 1 tablespoon olive oil, the vinegar, orange juice, parsley or chives, salt, and pepper in a screw-top jar. Cover and shake to combine. Place greens in self-sealing plastic bag; seal.
Transport vegetables and dressing in a picnic basket. Transport greens in an insulated cooler with ice packs.
To serve, toss together the beets, carrots and onions, greens, and dressing. Makes 6 side-dish servings.