Rinse and dry orange. With a vegetable peeler or small, sharp knife, pare colored part of peel from orange. Cut peels into very thin slivers. Squeeze juice from orange; measure 1/4 cup and save remaining for other uses.
Heat oil in a 10- to 12-inch wok or frying pan over medium-high heat. When hot, add orange peel, walnuts, and ginger and stir often just until peel begins to brown slightly at edges, about 2 minutes.
Stir in 1/4 cup orange juice and soy sauce. Add broccoli and stir occasionally until tender when pierced, about 5 minutes. Pour into a serving bowl.