To ensure even cooking, cut the chicken breasts in half lengthwise to create 4 total portions. Season both sides of chicken pieces with salt and pepper.
In a shallow bowl combine flour, ½ teaspoon salt, and ¼ teaspoon pepper. Dredge chicken breasts in the flour mixture shaking off excess, transfer to a plate.
In a medium bowl combine lemon zest, lemon juice, chicken broth, and capers. Set aside.
Heat a large 12-inch skillet over medium-high heat. Once hot add the olive oil. When the oil is hot and begins to shimmer, carefully add chicken to the pan. Cook until lightly browned and the thickest part of the meat reaches 160°F (61ºC), about 3 minutes per side. Remove and transfer to a plate.
Add lemon caper sauce and thyme sprigs to the same pan used to cook the chicken. Bring liquid to a simmer over medium heat. Scrape down the brown bits from the pan and whisk until dissolved into the liquid. Reduce by ⅓, about 2 to 3 minutes.
Return chicken to the pan and simmer over medium-low heat for 5 minutes, flipping halfway through. Transfer chicken to a serving dish. Remove and discard the thyme.
Over low heat, vigorously whisk in 2 tablespoons of butter into the sauce. Taste and season with more salt and pepper as desired. Add the chicken back and pour the sauce over top. Garnish with lemon wedges and parsley.
Add wine to the sauce: (Optional) Add ¼ cup of dry white wine to the sauce. Allow the sauce to simmer for an additional 1 minute before adding back the chicken to warm.