Rub beets with olive oil, and wrap in heavy-duty aluminum foil. Place packet on a baking sheet.
Bake at 350° for 45 minutes or until done; cool. Peel beets, and cut into thin strips.
Whisk together white wine vinegar and next 4 ingredients. Gradually whisk in vegetable and walnut oil until blended.
Arrange endive and radicchio on individual plates; top with apple slices and beet strips. Sprinkle with walnuts and cheese. Drizzle evenly with dressing, and serve immediately.