Prep the chicken. If the chicken breasts are very thick, slice them horizontally to create cutlets of even thickness (or pound them between plastic wrap). Season both sides generously with salt and pepper.
Heat your skillet. Place a large skillet over medium-high heat and add the olive oil. When the oil is shimmering but not smoking, it's ready.
Cook the chicken. Add the chicken breasts to the hot skillet and cook for about 5-6 minutes on each side until golden brown and cooked through (internal temperature of 165°F). Don't crowd the pan; work in batches if necessary. Transfer the chicken to a plate and cover loosely with foil to keep warm.
Start the sauce. In the same skillet, reduce heat to medium and add the butter. Once melted, add the minced garlic and chopped shallot. Sauté for 1-2 minutes until fragrant and translucent, but not browned. (Burned garlic = bitter sauce, and nobody wants that.)
Deglaze like a pro. Pour in the white wine (or additional broth) and use a wooden spoon to scrape up all those flavorful browned bits from the bottom of the pan. Let simmer for about 2 minutes until slightly reduced.
Build the flavor base. Add the chicken broth and Dijon mustard, whisking to combine until smooth. Let this simmer for about 3 minutes to reduce slightly.
Make it creamy. Reduce heat to low and slowly whisk in the heavy cream. Add the thyme leaves and bring to a gentle simmer (don't boil or the cream might curdle).
Finish the sauce. Stir in the parsley, lemon zest (if using), and lemon juice. Taste and adjust seasoning with more salt and pepper if needed.
Reunite chicken and sauce. Return the chicken to the skillet, spooning the sauce over the top. Let everything simmer together for 2-3 minutes until the chicken is warmed through and the sauce has thickened slightly.
Serve it up. Plate the chicken, generously spoon the sauce over the top, and garnish with additional fresh parsley.
