For the Herb Butter: Mix butter, shallot, herbs, lemon juice, salt, and pepper in medium bowl with rubber spatula until thoroughly combined. Form into 3-inch square on plastic wrap; wrap tightly and refrigerate until firm, about 1 hour.
For the Chicken: Adjust oven rack to lower-middle position; heat oven to 300 degrees. Add half of bread cubes to food processor and pulse until cubes are coarsely ground, about sixteen 1-second pulses. Transfer crumbs to large bowl and repeat with remaining bread cubes (you should have about 3 1/2 cups crumbs). Add 1/8 teaspoon salt and 1/8 teaspoon pepper to bread crumbs. Add oil and toss until crumbs are evenly coated. Spread crumbs on rimmed baking sheet and bake until golden brown and dry, about 25 minutes, stirring twice during baking time. Let cool to room temperature (you should have about 2 1/2 cups bread crumbs).
Prepare cutlets, following illustrations 1 and 2 below. Unwrap butter and cut into 4 rectangular pieces. Place chicken breast cut side up on work surface; season both sides with salt and pepper. Following illustration 3, place 1 piece of butter in center of bottom half of breast. Roll bottom edge of chicken over butter, then fold in sides and continue rolling to form neat, tight package, pressing on seam to seal. Repeat with remaining butter and chicken. Refrigerate chicken, uncovered, to allow edges to seal, about 1 hour.
Adjust oven rack to middle position and heat oven to 350 degrees. Place flour, eggs, and bread crumbs in separate pie plates or shallow dishes. Season flour with 1/4 teaspoon salt and 1/8 teaspoon pepper; season bread crumbs with 1/2 teaspoon salt and 1/4 teaspoon pepper. Add mustard to eggs and whisk to combine. Dredge 1 chicken breast in flour, shaking off excess, then coat with egg mixture, allowing excess to drip off (illustrations 4 and 5). Coat all sides of chicken breast with bread crumbs, pressing gently so that crumbs adhere (illustration 6). Place on wire rack set over rimmed baking sheet. Repeat flouring and breading of remaining chicken breasts.
Bake until instant-read thermometer inserted into center of chicken (from top) registers 160 degrees, 40 to 45 minutes. Let rest 5 minutes on wire rack before serving.