For the dressing: Whisk together the mayonnaise, lemon juice, Parmesan, olive oil, mustard, anchovy paste, pepper, salt and garlic in a large bowl. Set aside to let the flavors marry.
For the salad: Heat a large skillet over medium heat. Add the oil and heat another 30 seconds, then add the bread cubes. Cook, tossing often, until the bread cubes are beginning to brown and get crispy, about 6 minutes. Season with the salt and 2 tablespoons Parmesan and cook until the cubes are evenly golden and crispy, another 3 minutes. Remove to a paper towel-lined plate and reserve.
Add the baby kale to the bowl with the dressing. Toss gently to coat. Add the toasted bread and the remaining 1/4 cup Parmesan to the salad and toss once again.