Using a paring knife, remove peel, pith, and outer membranes from oranges. Carefully cut each segment away from membranes, collecting juice and segments in a bowl. Squeeze any remaining juice from membranes before discarding. Drain segments, and set aside, reserving juice for another use.
In a large bowl, combine lettuces and radicchio; toss with vinaigrette. Transfer to a serving platter, top with reserved orange segments, and sprinkle with cheese. Serve immediately.