1. Prepare the Chicken:
* If your chicken breasts are thick, place them one at a time between two pieces of plastic wrap or in a large zip-top bag. Using a meat mallet, gently pound them to an even thickness of about ½ inch. This ensures even cooking and tender results.
* In a shallow dish or plate, combine the all-purpose flour, 1 teaspoon salt, ½ teaspoon black pepper, and paprika. Mix well.
* Dredge each chicken breast in the flour mixture, ensuring it’s lightly coated on all sides. Shake off any excess flour. Set the coated chicken aside on a clean plate.

2. Sear the Chicken:
* In a large, heavy-bottomed skillet (cast iron works wonderfully), heat 2 tablespoons of olive oil and 2 tablespoons of butter over medium-high heat. Wait until the butter is melted and the oil is shimmering.
* Carefully place 2 or 3 chicken breasts in the hot skillet, being sure not to overcrowd the pan (work in batches if necessary).
* Sear the chicken for about 3-5 minutes per side, or until golden brown and cooked through. The internal temperature should reach 165°F (74°C).
* Remove the cooked chicken from the skillet and transfer it to a plate. Tent loosely with foil to keep warm. If you cooked in batches, add a little more oil/butter for the second batch if needed.

3. Sauté Aromatics and Mushrooms:
* To the same skillet (don’t wipe it clean – those browned bits are flavor!), add the remaining 1 tablespoon of butter and 1 tablespoon of olive oil. Heat over medium heat.
* Add the chopped onion and sauté for about 3-4 minutes, until softened and translucent.
* Add the sliced mushrooms to the skillet. Cook, stirring occasionally, for about 5-7 minutes, until they release their liquid and become nicely browned. Mushrooms absorb a lot of flavor, so allow them to caramelize slightly for the best taste.
* Stir in the minced garlic and cook for another minute until fragrant. Be careful not to burn the garlic.

4. Deglaze and Build the Sauce:
* Pour the dry sherry (or white wine) into the skillet. Use a wooden spoon or spatula to scrape up any browned bits (fond) from the bottom of the pan. Let the sherry simmer and reduce by about half, which should take about 1-2 minutes. This step adds incredible depth of flavor.
* Gradually pour in the chicken broth, stirring continuously. Bring the mixture to a gentle simmer.
* Reduce the heat to medium-low and stir in the heavy cream. Let the sauce simmer gently for 5-7 minutes, allowing it to thicken slightly. If you want a thicker sauce, you can mix 1 tablespoon of cornstarch with 2 tablespoons of cold water to make a slurry and slowly whisk it into the simmering sauce until desired consistency is reached.
* Stir in the ½ cup of grated Parmesan cheese until it’s melted and the sauce is smooth.
* Taste the sauce and adjust seasoning with salt and freshly ground black pepper as needed. If using, stir in the fresh lemon juice for a touch of brightness, or a pinch of nutmeg/thyme for warmth.

5. Combine and Finish:
* Return the seared chicken breasts (and any accumulated juices from the plate) to the skillet with the sauce.
* Nestle the chicken into the sauce, spooning some of the mushroom sauce over the top of each piece.
* Let everything simmer together gently for another 2-3 minutes, just enough for the chicken to reheat and absorb some of the sauce’s flavors.
* If desired, you can place the entire skillet (if oven-safe) under a preheated broiler for 1-2 minutes to get a little extra color on top, or sprinkle with more Parmesan and let it melt. Watch carefully to prevent burning.

6. Garnish and Serve:
* Remove the skillet from the heat.
* Garnish generously with freshly chopped parsley.
* Serve immediately, ensuring each portion of chicken gets a liberal amount of the glorious mushroom sauce.