Warm the olive oil in a large skillet over medium heat until it begins to shimmer. Generously season both sides of the chicken breasts with salt and pepper. Add the chicken to the skillet and leave it undisturbed until the underside is nicely browned, about 5 minutes.
Arrange the chicken in a single layer in a 6-quart or larger slow cooker, keeping the browned sides facing upward. Pour in the chicken broth and lemon juice, then add the smashed garlic cloves. Cover and cook on LOW for 3 to 4 hours, or until the chicken is tender, fully cooked, and reaches 165°F when checked with an instant-read thermometer.
While the chicken cooks, combine the room-temperature butter and flour in a small bowl. Using your fingers, work the flour thoroughly into the butter. Set this mixture aside until needed.
Once the chicken has finished cooking, transfer the breasts to individual plates or a serving platter.
Pour the liquid and garlic from the slow cooker into a small saucepan. Set it over medium-high heat and bring the mixture to a boil. Add the prepared butter-and-flour mixture, whisking continuously until it has completely blended into the liquid.
Continue cooking the sauce for 3 to 4 minutes, stirring frequently, until it thickens. Spoon or pour the finished lemon-garlic sauce over the chicken. Garnish with chopped fresh parsley, if desired, and serve.