Slice chicken breasts in half horizontally to make thin cutlets. Pound to about ½ inch thick.
Season with salt and pepper, then lightly coat with flour.
Heat a large pan over medium-high heat. Add olive oil and butter.
Cook chicken for about 2 minutes per side, or until lightly browned. Cook in batches if needed. Transfer to a plate.
Lower heat to medium. Add more olive oil, shallot, garlic, and capers. Cook for about a minute.
Add white wine and scrape up the browned bits from the pan.
Add chicken stock, lemon juice, and butter. Season with salt and pepper, then simmer for about 2 minutes.
Return chicken to the pan and cook until heated through, about 2 minutes.
Spoon the sauce over the chicken, then turn off the heat. Add lemon slices and fresh parsley before serving.