Combine lemon slices, sugar and smashed garlic in a medium bowl. Set aside.
Sprinkle chicken with half of the kosher salt and pepper. Heat a large skillet over medium-high heat. Add 2 tsp olive oil, swirl to coat. Add chicken to pan, cook 4 minutes per side, or until no longer pink. Set aside on a plate.
Add 1 tsp olive oil to skillet, swirl to coat. Turn heat to high. Add lemon mixture and cook 2 minute, until lemon is slightly charred, turning occasionally. Return to bowl.
Turn heat down to medium. Add 1 1/2 tsp butter to skillet and swirl until melted. Add shallot and minced garlic and saute for 1 minute. Sprinkle with flour and stir to combine. Add thyme sprigs and wine, scraping skillet to loosen browned bits. Bring to a boil, cook 3 minutes or until liquid is almost evaporated. Add chicken stock and remaining salt, simmer until reduced by half, about 3 minutes.
Remove pan from heat, discard herb sprigs, stir in remaining butter and capers, stirring until melted. Return chicken and accumulated juices to the pan and turn to coat in sauce. Top with lemon mixture. Sprinkle with chopped parsley and serve immediately.
Serve over hot cooked angel hair pasta or mashed potatoes.
Notes
Tips for Storing
Refrigerator: Store leftovers in an airtight container for up to 3 days. The sauce will thicken as it cools, which is completely normal and expected.
Reheating: Warm gently in a skillet over medium-low heat with a splash of chicken stock to loosen the sauce. The microwave works in a pinch, but it can make the chicken thighs rubbery if overheated.
Freezer: The chicken freezes well for up to 2 months. The butter-based sauce may separate slightly when thawed, but a quick whisk over low heat brings it back together.
Meal prep option: Sear the chicken and char the lemons ahead of time and refrigerate them separately. Build the pan sauce fresh right before serving for the best texture and flavor.
Storing sauce separately: If you are making this for a dinner party, keep the sauce in a separate container from the chicken overnight. This prevents the chicken from absorbing all the liquid during storage.