Put on a pair of rubber gloves and slice the beets into thin rounds on a mandoline. Stack the rounds and then slice them into thin strips (aka julienne). Place the julienned beets on a paper-towel-lined plate. Set aside.
In a medium bowl, whisk together the olive oil, orange juice, balsamic vinegar, honey, and salt and pepper to taste. Set aside.
Remove 8 leaves from each endive, and reserve. Slice the remaining endives 1/2 inch thick. Combine the sliced endives, radishes and avocado in a large bowl.
Season with salt and pepper. Add half of the dressing and toss to coat.
In a second bowl, add the julienned beets and pour over the remaining dressing. Toss to coat.
Set up 4 salad plates. Arrange 4 of the reserved endive leaves on each plate, with the stem ends pointed toward the center. Top each leaf with some of the avocado-endive mixture and then with some of the beets. Serve immediately.