Pat the chicken breasts dry with paper towels, then season generously with salt and pepper.
Heat a skillet over medium-high heat and add a tablespoon of olive oil. Once shimmering, sear the chicken for about 2-3 minutes per side until golden brown. Transfer the seared chicken to your slow cooker.
In the same skillet, add minced garlic and cook until fragrant, about 30 seconds. Pour in chicken broth, lemon juice, and a splash of white wine if you like. Scrape up any browned bits from the pan and bring to a gentle simmer.
Pour the garlic-infused broth mixture over the chicken in the slow cooker. Add the rinsed capers evenly across the top. Cover and cook on low for 4-5 hours until the chicken is tender and cooked through.
Once cooked, carefully remove the chicken breasts using tongs and set aside on a plate. Turn the slow cooker to high. In a small bowl, whisk together flour and a bit of water to create a slurry, then stir into the sauce to thicken. Let simmer uncovered for 10-15 minutes until glossy and slightly reduced.
While the sauce thickens, allow the chicken to rest for a few minutes. Slice the chicken if desired, then spoon the lemony caper sauce generously over each piece. Finish by stirring in the butter for a silky, rich finish.
Serve the chicken hot over pasta, rice, or with crusty bread, drizzling with extra sauce and garnishing with fresh herbs if you like.
