How to Make It
Preheat oven to 425°F. Toss bread, oil, and ¼ teaspoon each salt and pepper on a rimmed baking sheet. Spread in an even layer and bake until golden brown and crispy, about 10 minutes. Transfer croutons to a bowl and let cool for 10 minutes.
Meanwhile, whisk mayonnaise, ½ cup water, grated cheese, garlic, anchovies, mustard, lemon juice, and remaining ¼ teaspoon each salt and pepper in a large bowl. Add Brussels sprouts, chicken, and kale; toss gently to coat.
Transfer salad to a platter. Top with croutons and shaved cheese.