Pat the chicken breasts dry with paper towels, then slice each into thinner cutlets for even cooking and tenderness.
Lightly season the chicken with salt and pepper, then place them in the slow cooker, spreading them out to ensure even contact with the sauce ingredients.
In a mixing bowl, whisk together lemon juice, white wine, minced garlic, and chicken broth until well combined. Pour this flavorful mixture evenly over the chicken in the slow cooker.
Cover the slow cooker and cook on low for 3 to 4 hours, until the chicken is tender and easily shredded with a fork. The kitchen will fill with a bright, garlicky aroma.
About 15 minutes before serving, carefully transfer the cooked chicken to a plate and set aside. Turn the slow cooker to high to prepare for finishing the sauce.
Whisk in a tablespoon of butter into the sauce to add shine and richness, stirring until melted and glossy. Return the chicken to the slow cooker to reheat in the sauce for about 5 minutes.
Gently nestle the chicken back into the sauce, allowing it to soak up the bright, tangy flavors and become perfectly tender. The sauce should shimmer and slightly thicken, with a fragrant aroma.
Serve the chicken hot, spooning generous amounts of the lemony caper sauce over each piece. Garnish with fresh parsley if desired for a burst of color and freshness.