Heat 1 tablespoon butter in either a cast iron skillet or a nonstick skillet. Season chicken with salt and pepper and add to skillet. Cook until golden brown, about 3 minutes per side. Place in 6 quart crock pot.
Cut remaining 4 tablespoons of butter into 4 pieces and place a piece over each chicken breast. Add basil, red pepper flakes, lemon juice, and lemon zest. Cover crock pot and cook on LOW for 3 1/2 hours or on HIGH for just under 2 hours.
Add 1 tablespoon of the 1 cup of half-and-half to a small bowl and whisk in the cornstarch. Whisk into the rest of the half-and-half. Whisk in chicken base and add the capers. Add half-and-half mixture to crock pot. Mix it in and cover and cook on HIGH for 1 more hour.
