Pound the chicken breasts to an even 1-inch thickness and season with salt and pepper.
Dredge the chicken in flour, shaking off any excess.
In a skillet over medium-high heat, melt 1 tablespoon of butter and add olive oil. Pan-fry the chicken for 4 to 5 minutes on each side until golden brown.
Once cooked, remove the chicken and set it aside.
Melt remaining butter in the skillet, add minced garlic, and sauté for about 30 seconds until fragrant.
Pour in the chicken broth and scrape the browned bits from the pan.
Add in the heavy cream and bring to a gentle boil, then stir in capers, lemon juice, and parsley.
Adjust seasoning with salt and pepper and simmer until thickened, adding cornstarch slurry if desired.
Return chicken to the pan and spoon sauce over. Simmer for 1 to 2 minutes until heated through.
Garnish with parsley and lemon slices, and serve hot over pasta, rice, or mashed potatoes.