Prepare the chicken by slicing each breast in half horizontally, creating 4 pieces in total. Pat dry with paper towels. Season both sides of the pieces with Italian seasoning, sea salt, and freshly cracked black pepper to taste.
Prepare the lemon by finely grating the zest from the large lemon and place it in a small bowl. Halve the lemon crosswise. Juice one half until you have 2 tablespoons of fresh juice, and add it to the bowl. Cut 4-5 (¼-inch-thick) slices from the remaining half of the lemon. Side Note: Remove the seeds and reserve the slices.
Make the sauce mixture by whisking ¾ cup cold chicken broth, 4 finely grated garlic cloves, and the flour into the lemon juice mixture. Season with sea salt and freshly cracked black pepper to taste and whisk until no lumps remain.
Cook the chicken by heating the olive oil in a large nonstick skillet over medium-high heat until just smoking. Add the chicken in a single layer and cook until well-browned, 3 to 4 minutes per side (the chicken might not be cooked through).
Reduce the heat to medium-low. Re-whisk the lemon juice mixture, pour it into the skillet, and add the reserved 4-5 lemon slices. Simmer until the sauce thickens and the chicken is cooked through, flipping halfway through, 3 to 4 minutes total.
Turn off the heat. Transfer the chicken to a plate and loosely cover with a foil tent.
Finish the sauce by adding 4 tablespoons of cold butter to the pan and whisking until melted and combined. Taste the sauce and season with sea salt and freshly black pepper as needed. Stir in the finely chopped parsley.
Return the chicken and any accumulated juice to the pan; flip the chicken to coat with sauce.