With a sharp kitchen knife, cut each chicken breast lengthwise in two thin cutlets. Take each scallop and pound it gently to thin it out a bit further. Trim off any gristle or rough edges.
Dredge each chicken cutlet in flour, seasoned generously with salt and pepper, and then dip it in the egg to coat lightly.
In a large skillet, over moderate heat, brown the chicken cutlets on both sides in olive oil and set aside.
Add a splash of white wine to the skillet and let it evaporate almost entirely, then add the chicken broth.
Let everything simmer for a few minutes so the flavors meld. Add the flour, kneaded together with the butter into a small ball and whisk it into the liquid, which should thicken into a sauce like consistency.
Add the chicken cutlets back into the skillet, along with the lemon slices, and let everything simmer in the sauce for a minute or two to warm up. If need be, add more broth or water so you have ample sauce for napping the cutlets, but they shouldn't 'swim' in it.
At the last moment, taste and adjust the sauce for seasonings, then drizzle in the lemon juice, to taste.
Serve the chicken cutlets hot, napped with the sauce and garnished with the lemon slices and minced parsley.