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 kale with 1/4 cup 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 apple slices and nuts 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 (toss apples or pears with a little lemon juice to prevent browning).
Greens and Pear Salad: Replace the apples with 2 large or 3 small sliced and cored pears, the kale with radicchio, and the cheddar cheese with 6 oz. crumbled feta cheese.
Greens and Orange Salad: Replace apples with segments from 3 large oranges and sub in arugula for the kale. Replace cheddar with 1 cup shaved pecorino or parmesan cheese