In a food processor, pulse bread, Parmesan, parsley, and 1/4 teaspoon each salt and pepper until fine crumbs form. Transfer mixture to one
shallow bowl; place egg in another.
One at a time, dip each chicken cutlet into egg; let excess drip off, then dip into breadcrumb mixture.
Press on crumbs firmly.
Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Cook cutlets, two at a time, adding remaining tablespoon oil for second batch, until cooked through, 2 to 3 minutes per side.
Wipe skillet clean. Add lemon juice and 1/4 cup water; cook over low until steaming, 1 minute. Add mustard. Remove pan from heat, and stir in butter until combined. Serve cutlets with sauce.
When adding butter to the lemon-Dijon sauce, be sure to stir vigorously with a wooden spoon or a heatproof spatula.