Preheat oven to 375°. Scrub beets, pat dry, rub with 1 tbsp. olive oil, and sprinkle generously with salt. Put beets on a foil-covered baking sheet and bake until tender when pierced, about 1 hour. Refrigerate uncovered until cool enough to handle, about 30 minutes. Cut off roots and stems, then rub beets with paper towels to remove skin; discard skin.
Cut beets in half lengthwise, then slice into half-moons about 1/4 in. thick; set aside. Cut ends off oranges, then cut away peel and outer membrane in wide strips, following the curve of the fruit with knife. Discard peel. Working over a bowl to catch juices, cut oranges between inner membranes and fruit to release segments into bowl; squeeze juice from membranes into bowl and discard membrane.
Finely zest limes into another bowl. Juice limes and add juice to zest. Add shallots, vinegar, and juice from oranges. Whisking constantly, slowly drizzle remaining 4 tbsp. olive oil into bowl. Add beets, toss to coat, and season to taste with salt and pepper. Cover bowls airtight and let stand at room temperature at least 15 minutes and up to 3 hours.
Arrange arugula on a platter. Sprinkle with cilantro leaves. Pour beets and dressing over arugula and scatter with orange segments and cheese.
Note: Nutritional analysis is per serving.