Cut off ends of oranges; using a paring knife, remove peel, pith, and outer membranes, following the curve of the fruit. Working over a bowl to catch the juices, cut between sections to remove whole segments; add segments to a large bowl. Squeeze juice from membranes into a bowl; measure out 3 tablespoons juice; reserve remaining juice for another use.

In a small bowl, whisk together reserved orange juice, vinegar, and olive oil; season with salt and pepper. To bowl with oranges, add lettuce, avocado, and radishes. Toss with dressing, and serve immediately.