Make herb butter: Mix the butter, shallot, herbs, lemon juice, salt, and pepper together in a medium bowl. Place mixture on plastic wrap, and shape it into a 3 inch square, wrap it tightly in plastic wrap, and refrigerate until firm, about an hour.
Preheat Oven: Adjust oven rack to the lower middle position and heat to 300 degrees F.
Make bread crumbs (Or skip and use already made breadcrumbs): Pulse the bread in a food processor to coarse crumbs, about 8 pulses. Toss the breadcrumbs with the oil and ⅛ teaspoon each salt and pepper until coated. Bake the crumbs on a rimmed baking sheet until golden brown and dry, about 25 minutes, stirring twice during the baking time. Let the crumbs cool to room temperature.
Prep Chicken: Cut the herb butter into 4 rectangular pieces. Use paper towels to pat the chicken dry, then season with salt and pepper. Cut chicken breasts in half, but leave in tact, so do not cut all the way through. Spread apart, and take chicken and place the breast cut side up on the work surface. Pound to flatten into an even thickness. Pound thinner at the edges to make sealing the chicken together easier. Then place 1 piece of the prepared butter into the center of the bottom half of each breast. Roll the bottom edge of the chicken over the butter, then fold in the sides and continue rolling to form a neat, little packet. Press the chicken seams together to help it seal. Then place it seam side down on a plate, and refrigerate the chicken, uncovered, for about 1 hour.
Prep Oven and Baking Dish: Adjust oven rack to the middle position and heat the oven to 350 degrees F. Place a wire rack over a rimmed baking dish, and set aside.
Prepare Dredge Station: Place the flour, eggs, and bread crumbs in three separate shallow dishes. Season the flour with ¼ teaspoon salt and ⅛ teaspoon pepper. Season the breadcrumbs with ½ teaspoon salt and ¼ teaspoon pepper. Crack and whisk eggs with mustard.
Dredge Chicken: Dredge each stuffed chicken breast in flour, shaking off the excess, then coat with the egg mixture, allowing the excess to drip off, then coat the chicken with the breadcrumbs, pressing gently so that the crumbs adhere.
Place dredged chicken on a wire rack set over the rimmed baking sheet.
Bake for 40-45 minutes, until an instant read thermometer inserted into the center of the chicken registers 160 degrees F. Let the chicken rest for 5 minutes on the wire rack before serving.