In a shallow bowl, combine the flour and parmesan cheese. Season the chicken with salt and pepper; dredge in the flour mixture, ensuring each piece is coated evenly. Shake off any excess and set aside.
Heat 1 tablespoon of olive oil and 2 teaspoons of butter in a large skillet over medium-high heat until the butter has melted and the pan is hot. Carefully add the dredged chicken to the skillet. Fry each side until it's golden brown and cooked through, which should take about 3 to 4 minutes per side, depending on the thickness of your chicken. Transfer it to a warm plate.
Add the minced garlic to the hot oil in the pan (if needed, spray with a light coating of oil) and fry until fragrant, about 1 minute.
Reduce the heat to low-medium, then add the chicken broth and cream to the pan. Stir to combine everything while scraping up any bits from the bottom of the skillet; these add extra flavor to your sauce.
Bring the sauce to a gentle simmer. Season with additional salt and pepper to taste. Now, add in the additional parmesan cheese and capers, stirring gently. Let it continue to cook for about 2 minutes, allowing the sauce to thicken slightly.
Pour in the fresh lemon juice and allow the sauce to simmer for an additional minute to balance the flavors.
Return the fried chicken to the skillet, allowing it to soak up the sauce for about a minute.
Once everything is combined beautifully, serve immediately. I recommend plating the chicken over pasta, drained zucchini noodles, or alongside steamed vegetables.
