Rinse the chicken and pat it dry with paper towels. Place it in a glass baking dish or a large resealable plastic bag. Add the buttermilk. Cover and refrigerate for at least 1 hour and up to 4 hours.
Heat oven to 400° F.
Place the bread crumbs, Parmesan, salt, and pepper on a plate. Working with 1 piece of chicken at a time, shake it to remove any excess buttermilk and dredge it in the bread-crumb mixture, turning to coat both sides.
Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add half the chicken. Cook until golden, about 3 minutes per side. Transfer to a baking sheet. Wipe out skillet. Repeat with the remaining oil and chicken. Bake until cooked through, about 10 minutes.
Tip: If you (or the kids) prefer a higher proportion of bread-crumb coating to meat, use chicken tenders in place of chicken breasts.