Flatten each chicken breast to about 1/2-inch thickness using a meat mallet or rolling pin. Pat completely dry with paper towels. Mix the flour, salt, and pepper on a plate. Dredge each breast lightly in the seasoned flour and shake off any excess.

Heat 3 tbsp of butter with the olive oil in a large skillet over medium-high heat until shimmering. Add the chicken and cook 4 to 5 minutes per side until deep golden and cooked through to 165 degrees F. Cook in batches if needed to avoid crowding the pan. Transfer to a plate and cover loosely with foil to keep warm.

Reduce heat to medium. Add the remaining 3 tbsp of butter to the same skillet. Once melted, add the minced garlic and cook for 30 to 60 seconds until fragrant. Do not let it brown.

Pour in the white wine and chicken stock. Use a wooden spoon to scrape up all the browned bits from the bottom of the pan. Let the liquid bubble and reduce for 2 to 3 minutes.

Stir in the heavy cream and fresh parsley. Simmer gently for 2 to 4 minutes until the sauce thickens enough to coat the back of a spoon. Taste and adjust with salt and pepper.

Return the chicken to the skillet and spoon the sauce generously over each breast. Let everything warm through for 1 to 2 minutes. Plate immediately and serve with lemon wedges on the side.