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.
In a bowl, toss radicchio with 3 tbsp. vinaigrette until leaves are coated evenly. Cover and leave at room temperature, at least 30 minutes and up to 4 hours.
Just before serving, add butter lettuce, romaine, and pear slices and toss gently. Scatter with cheese.
Make ahead: Dressing, up to 2 weeks ahead, chilled; other ingredients can be prepped several hours ahead and chilled separately (toss pears with a little lemon juice to prevent browning).