Preheat oven to 200°F (95°C). Place a wire rack inside a baking sheet; set aside.

In a shallow dish or plate, stir together flour, 1 ½ teaspoons kosher salt, and ¾ teaspoon pepper; set aside. Set aside a large plate to transfer chicken to after breading.

Working with one chicken breast at a time, place chicken breast on a cutting board. Hold it in place with the palm of your non-cutting hand, and use a sharp knife to slice chicken horizontally into two even cutlets. Repeat process with remaining chicken breast. Cover each breast half with plastic wrap, and use a meat mallet or rolling pin to pound to ⅓-inch thickness.

Place 2 chicken halves in prepared flour mixture, and turn to coat; tap off excess flour mixture, and transfer chicken to the large plate. Repeat process with remaining chicken halves.

In a large stainless steel, cast iron, or carbon steel skillet, heat oil over medium-high until shimmering. Working in batches, add two chicken pieces to hot oil; cook, undisturbed, until golden brown on both sides, 4 to 5 minutes total. Repeat process with remaining 2 chicken pieces. Place browned chicken halves on prepared baking sheet, and place in warm oven (do not wipe out skillet).

While chicken is resting in oven, heat the same skillet over medium-high. Add wine; cook, stirring occasionally and scraping up browned bits from bottom of skillet, until reduced to about ¼ cup, 3 to 4 minutes. Add stock and cook, stirring constantly, until mixture has reduced to about ⅓ cup, 3 to 4 minutes. Reduce heat to medium-low. Add garlic, butter, lemon juice, 1 tablespoon of parsley, 1 tablespoon of chives, 2 teaspoons tarragon (if using), remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper; cook, swirling and stirring constantly, until butter is completely melted, about 1 minute. Spoon sauce over chicken, and garnish with remaining chives, parsley, and additional tarragon (if using), lemon zest, and flaky salt. Serve immediately.