Pound out chicken breast halves between 2 sheets of plastic wrap or in a resealable bag to 1/3- to 1/2-inch thickness.

Mix eggs, parsley, and Dijon mustard together in a large bowl. Pour panko into a separate bowl. Place chicken in the egg mixture, then dip in panko.

Heat olive oil over medium-high heat in a large, nonstick skillet. Add chicken breasts and cook until no longer pink in the centers and juices run clear, about 4 minutes per side. Transfer to a plate and keep warm.

Whisk chicken broth, maple syrup, and 2 tablespoons plus 1 teaspoon coarse-grain mustard together in a bowl. Pour into the same skillet used to cook chicken and bring to a boil. Cook sauce until reduced to about 3/4 cup, about 4 minutes. Stir in butter until melted. Spoon sauce over chicken and serve.