First things first. Grab three shallow bowls and arrange them in a triangle on your workspace. Pour the flour into one bowl, your beaten eggs into the second, and mix panko breadcrumbs with grated Parmigiano-Reggiano in the third. This layout makes your breading process smooth and efficient, like a well-choreographed dance.

Prepare your chicken cutlets by seasoning them generously with salt and freshly ground black pepper on both sides. Next, take one cutlet at a time. Dredge it in the flour, ensuring you cover all surfaces. Shake off any excess flour before moving it to the egg dish. Use your other hand to coat the cutlet fully in the beaten egg, allowing any excess to drip back into the bowl.

Now, it’s time to give those cutlets a crispy coat. Transfer the egg-coated cutlet to the bowl with the panko and cheese mixture. With one hand, sprinkle the breadcrumb mixture over the top while gently pressing it down to make sure it adheres well. Flip it over and do the same on the other side. Transfer the breaded cutlet to a parchment-lined baking sheet. Repeat this process for all the chicken cutlets.

In a large skillet, add vegetable oil until it's about 1/4 inch deep. Heat the oil over high heat until it shimmers—this means it's nearly ready for frying. If you have a thermometer handy, you’re aiming for about 375°F (191°C). It’s crucial not to skip this step; the oil must be hot enough to create a crisp coating without absorbing too much oil.

Carefully place the breaded cutlets into the hot oil, laying them down away from you to prevent any splashes. Fry for about three minutes, gently swirling the pan to ensure even browning. After three minutes, flip the cutlets over—look for that beautiful golden-brown crust. Fry them for an additional three minutes. Once done, transfer to a plate lined with paper towels to drain excess oil, and season immediately with salt.

Now, don’t waste that delicious frying oil! Drain off all but one tablespoon from the skillet. Add your dry white wine to the hot pan, and bring it to a vigorous boil over medium-high heat. Cook for about two minutes, until most of the raw alcohol smell is gone. Toss in the drained capers and butter, whisking continuously until the butter melts and the sauce begins to thicken.

Lower your heat to medium, and continue whisking until you have a smooth and glossy sauce. If it looks too thick, you can add a tablespoon of cold water. Once you’re happy with the consistency, stir in the freshly squeezed lemon juice, minced parsley, and season with salt and pepper if needed.

Arrange the crispy chicken cutlets on a serving platter and generously drizzle the warm butter sauce all over. Serve immediately for the best experience. Your guests will be impressed, and you might want to take a bow!