Pound and Season the Cutlets: Place each chicken cutlet between two sheets of plastic wrap and pound with a meat mallet or the bottom of a heavy skillet to an even 0.6cm / 0.25 inch thickness. Season both sides with kosher salt and freshly ground black pepper. Uniform thickness is what guarantees even cooking throughout the saute - any thick patch will still be raw when the thinner edges are done.
Dredge in Flour: Spread the all-purpose flour in a shallow dish. Press each cutlet into the flour on both sides, then shake off the excess firmly - you want a thin, even coating, not a crust. Set the dredged cutlets on a clean plate and let them rest for 2 minutes before they go into the pan; this helps the flour adhere and prevents it from falling off into the oil and burning.
Sear the First Batch: Heat a large stainless steel skillet over medium-high heat and add 2 tablespoons of olive oil. Once the oil shimmers and just begins to smoke at the edges, lay two cutlets in the pan without crowding. Sear for exactly 3 minutes per side at medium-high heat until the surface is deep golden and releases cleanly from the pan. Transfer to a plate and tent loosely with foil.
Sear the Second Batch: Add the remaining 1 tablespoon of olive oil and repeat with the second pair of cutlets, again 3 minutes per side. The fond that has built up in the pan is flavor; do not wipe it out. Remove the cooked cutlets and confirm an internal temperature of 74°C / 165°F with an instant-read thermometer before setting aside.
Build the Sauce Base: Reduce the heat to medium and add 1 tablespoon of butter to the same pan. Once it foams, add the minced garlic and cook for 45 seconds, stirring constantly, until fragrant and pale gold. Add the dry white wine and use a wooden spoon to scrape up the fond from the bottom of the pan - the liquid will sizzle aggressively on contact and the browned bits will dissolve into the sauce. Let the wine reduce for 2 minutes until the sharp alcohol smell mellows.
Add Lemon and Broth: Pour in the chicken broth and lemon juice, raise the heat to medium-high, and let the sauce simmer for 3 minutes. It should reduce by roughly one-third and take on a slightly glossy appearance as it concentrates. Stir in the rinsed capers.
Finish with Butter and Return Chicken: Remove the pan from the heat. Swirl in the remaining 1 tablespoon of cold butter until it melts into the sauce and creates an emulsified, lightly creamy consistency - this is the monte al burro technique, and pulling the pan off the burner is what prevents the butter from breaking and turning greasy. Nestle the seared cutlets back into the sauce and spoon the liquid over them. Return the pan to low heat for 1 minute at 90°C / 195°F just to warm the chicken through without overcooking it.
Plate and Garnish: Transfer the cutlets to plates or a serving platter and spoon the lemon-caper sauce generously over the top. Scatter the chopped flat-leaf parsley over everything and lay lemon rounds alongside.
