Preheat oven to 350°F. Trim beets and wrap separately in foil; crimp edges. Roast on a rimmed baking sheet until tender, 45 minutes to 1 hour.
Cut a thick slice off top and bottom of each orange. Use a sharp knife to cut away peel and pith from fruit, slicing from top to bottom. Working over a bowl to catch juices, cut between membranes and remove orange segments. Drain on paper towels. Reserve juices.
Cut fennel bulbs into quarters, remove core and pull apart layers. Thinly slice each layer and transfer to bowl of collected orange juice; toss.
Remove beets from oven and carefully pull open foil. (Be careful: Hot steam will escape.) When cool enough to handle, hold under cold running water and rub off skins. Halve beets; cut each half into 1/4-inch-thick slices.
Make vinaigrette: Whisk vinegar, lemon juice, salt, pepper and mustard. Slowly add both oils, whisking constantly.
Assemble salad: Drain juice from fennel and toss fennel with 2 Tbsp. vinaigrette. Arrange fennel on salad plates and place oranges and beets on top. Drizzle remaining vinaigrette over salad.