Slice the chicken breasts in half horizontally to make 8 thin cutlets. Season with salt and pepper and dredge in flour. Heat the butter in a large skillet over high heat. Add the chicken in batches and cook until browned and cooked through, 3 to 4 minutes per side. Transfer to a plate.
Add the onion, thyme and 1/2 teaspoon salt to the skillet. Cook until the onion is golden, about 5 minutes. Add the chicken broth and cook until slightly reduced, 3 to 4 minutes. Reduce the heat to medium low.
Meanwhile, mix the egg yolks, pickles, pickle juice, scallions and parsley in a bowl. Stir into the broth and cook, stirring, until the sauce begins to thicken, about 1 minute. Add the chicken to the skillet and cook until the sauce thickens and the chicken is heated through, about 5 minutes (do not boil). Serve with bread and pickled beets, if desired.