First, I mix together the flour and the finely grated Parmesan in a shallow bowl, then season the chicken fillets with salt and pepper. I dredge each piece in this mixture, shaking off the excess so the coating is light but flavorful.
Next, I heat the olive oil and butter in a large skillet over medium-high heat. Once the butter melts and the pan is hot, I fry the chicken for about 3–4 minutes on each side (depending on thickness) until golden and cooked through, then transfer it to a warm plate.
In the same pan, I add the minced garlic (spraying a little extra oil if needed) and sauté for around a minute until it becomes fragrant but not brown.
Then, I lower the heat to medium, pour in the chicken broth and the half-and-half (or cream), stirring well as the sauce comes together.
Once the sauce is warm, I stir in the freshly grated Parmesan and the capers. If the consistency feels too thin, I whisk in the cornstarch slurry right into the center of the pan — it thickens almost immediately.
After that, I pour in the lemon juice and let the sauce simmer for another minute, then gently add the chicken back into the pan so it can soak up all that creamy, tangy goodness.
Finally, I spoon that luxurious sauce over the chicken. I usually garnish with extra capers, lemon slices, and fresh parsley before serving — it just makes everything look (and taste) more inviting.