Sear the chicken. Heat the olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Sear the chicken until golden on both sides and the thickest part reaches 165°F. Cooking time will vary depending on thickness. Transfer the cooked chicken to a plate and set aside.

Sauté the aromatics. Reduce the skillet heat to medium. Add the diced shallot to the remaining pan drippings and cook for 2–3 minutes, until softened. Add the minced garlic and cook for another minute, until fragrant.

Deglaze the skillet. Pour in the white wine, scraping up the browned bits from the bottom of the skillet. Simmer for approximately 2 minutes, until the wine reduces by half.

Make the creamy sauce. Add the chicken broth and simmer briefly to reduce slightly. Stir in the heavy cream, Dijon mustard, and drained capers. Simmer gently until the sauce thickens slightly. Adjust the simmering time as needed rather than adding a thickener.

Add the resting juices. Pour any accumulated juices from the cooked chicken into the sauce and stir to combine. Keep the chicken aside until serving.

Finish the sauce. Remove the skillet from the heat. Stir in the fresh lemon juice, then gradually whisk in the remaining 2 tablespoons of cold butter until the sauce is smooth and glossy. Taste and adjust the salt and pepper if needed. Do not return the sauce to a simmer.

Serve. Place the cooked chicken on serving plates and spoon the creamy lemon caper sauce over the top. Serve immediately.