Cut broccoli into 1x2" pieces. Cut zucchini into 1/2"-thick slices. Cut each green onion into 2" pieces (if root ends are too thick, cut each lengthwise in half).
In 8-quart Dutch oven over medium heat, in 1/3 cup hot salad oil, cook green onions, stirring frequently, until tender, about 3 minutes. With rubber spatula, stir in broccoli and 1 1/4 teaspoon salt until broccoli is coated with oil. Add water; cover Dutch over and cook broccoli 5-8 minutes, stirring occasionally, until tender-crisp. Remove vegetables and liquid to bowl.
In same Dutch oven over medium heat, in 1/4 cup hot salad oil, cook zucchini, basil, and 1 teaspoon salt, stirring occasionally, until zucchini is tender-crisp. Return broccoli mixture to Dutch oven; add soy sauce; mix well.
Line large platter with lettuce leaves. Spoon vegetables onto lettuce-leaf-lined platter. Serve hot or refrigerate to serve cold later.