How to Make It
Make dressing: Combine apple cider vinegar, ginger, honey, mustard, salt, and pepper in a jar with a tight-fitting lid. Shake until honey dissolves and ingredients are well combined. Add oil and shake vigorously to mix well.
Peel oranges and slice thinly crosswise. Just before serving, toss arugula with 3 tbsp. vinaigrette in a bowl until leaves are coated evenly. Add butter lettuce, romaine, and orange segments and toss gently. Scatter with shaved cheese.
Make ahead: Dressing, up to 2 weeks ahead, chilled; other ingredients can be prepped several hours ahead and chilled separately.