Heat 1 tablespoon oil in a medium nonstick skillet set over medium heat. Add the bread pieces and cook, stirring, until lightly golden and a bit crisp, about 5 minutes. Remove from the heat and set aside to cool in the skillet. They will continue to crisp up as they cool.
Whisk together the buttermilk, 2 tablespoons of the Parmesan, the sour cream, remaining 1 tablespoon olive oil, the lemon juice and garlic in a small bowl and until smooth. Season with pepper and 1/4 teaspoon salt.
Toss together the chicken, kale and croutons with the dressing in a large bowl until well coated. Divide among 4 plates, sprinkle evenly with the remaining 2 tablespoons Parmesan and serve with a lemon wedge.