Add the six chicken breasts to the insert of a 6 or 8 quart slow cooker.
In a glass measuring cup, add the flour to the melted butter. Microwave on high power for one minute. Stir. Repeat this process two more times. The butter and flour mixture will bubble up each time.
To the microwaved butter and flour mixture (roux) whisk in the can of chicken broth until well combined. Then stir in the pressed or minced garlic cloves, salt, capers and lemon juice. Gently pour the sauce over the chicken.
Cook the chicken and sauce on high power for 3-4 hours or on low power for 6-8 hours. Be sure and check the internal temperature of the chicken to make sure it is a minimum of 165 degrees F.
If you'd like to add some color to your chicken, you can reverse sear it in a non-stick skillet over medium high heat. But this step is completely optional.
Once you've removed the chicken from the sauce, whisk in the heavy cream and parsley. Return the chicken to the slow cooker, cover and keep on low to reheat the sauce and chicken.
Cook a box of angel hair pasta in salted water according to package directions. Drain the pasta and then mix it into the pot with the chicken and sauce.
Garnish with fresh lemon slices, more parsley and capers. Serve immediately.