Place each chicken breast half between sheets of plastic wrap and hammer with a mallet, or the bottom of a small frying pan, to about a 1/4 inch thickness.
Add oil to a saute pan until the bottom is just covered, and set it over a medium-high flame until quite hot. Season chicken with salt and pepper and lightly dredge in the flour, shaking off any excess.
When the pan is ready, add 1 tablespoon of butter and swirl to spread it evenly, and immediately add 2-3 pieces of chicken. Cook about 2 minutes on the first side, turning when the bottom is nicely browned. About 1-2 minutes on the second side should do it. Transfer the cooked pieces to a warm platter, add more oil and butter to the pan, and repeat the process with a second (and third) batch of chicken, as necessary.
Drain any excess oil from the pan, deglaze with a splash of white wine, and cook until almost fully reduced.
Toss in a fresh tablespoon of butter and the minced shallots, and cook for about a minute.
Add the capers, lemon juice, and chicken stock. Cook until slightly thickened.
Remove the pan from the heat, swirl in another 1-2 tablespoon of butter and some chopped parsley. Check for seasoning, and pour over the chicken breasts to serve.