Pulse bread, Parmesan, parsley, salt, and pepper in a food processor until fine crumbs form.

Transfer crumb mixture to a shallow bowl; place beaten egg in another shallow bowl.

Dip each chicken cutlet into the egg, allowing excess to drip off.

Dip the egg-coated cutlet into the breadcrumb mixture, pressing firmly to adhere.

Heat 1 tablespoon of olive oil in a large nonstick skillet over medium heat.

Cook cutlets in batches (two at a time), adding the remaining oil for the second batch, until cooked through (2-3 minutes per side).

Wipe the skillet clean.

Add lemon juice and 1/4 cup of water to the skillet; cook over low heat until steaming (1 minute).

Stir in Dijon mustard.

Remove the pan from the heat and stir in cold butter until combined and the sauce is emulsified.

Serve the Parmesan-crusted chicken cutlets with the lemon-butter sauce.