How to Make It
Carefully slice a pocket into the side of each chicken breast and season with salt and pepper. In a small bowl, blend the butter with the dill and chives and season with salt and pepper. Spoon half of the herb butter into the pockets and pinch closed.
Dredge the chicken in flour, tapping off the excess. Put the eggs and bread crumbs in 2 separate shallow bowls and season with salt and pepper. Dip the chicken in the eggs and then in the bread crumbs, sealing the pockets; dab any open spots with egg and sprinkle with bread crumbs to seal. Place the chicken on a baking sheet and freeze for 5 minutes.
In a large skillet, heat 1/2 inch of olive oil until shimmering. Add the chicken and fry over moderate heat, turning once or twice, until golden and cooked through, about 8 minutes.
In a large bowl, whisk the 2 tablespoons of olive oil with the lemon juice and season with salt and pepper. Add the arugula and toss. Transfer the chicken to plates and top with the remaining herb butter. Serve the arugula salad alongside.