To prepare the chicken, start at the thicker end of the chicken breast and use a sharp knife to cut the chicken breast in half into two thinner cutlets. Trim any excess fat and season both sides with kosher salt and freshly ground black pepper.
Heat a large skillet over medium-high heat. Add 1/2 tablespoon each of butter and olive oil until the butter melts. Place 3 chicken breasts in the pan and cook undisturbed for 3-4 minutes until one side is nicely browned and cooking through. Flip with tongs and cook for an additional 3-4 minutes or until the thickest part reaches 160°F. Transfer to a plate, tent with aluminum foil, and let it rest.
Using paper towels, wipe the pan and add 1/2 tablespoon of butter along with the remaining olive oil. Repeat the process with the remaining three chicken breasts and transfer them to the plate with the previously cooked chicken.
Wipe the pan once more and melt 1 tablespoon of butter over medium heat. Add mushrooms and leeks to the pan with a pinch of kosher salt. Cook, stirring occasionally, for approximately 5 minutes or until the mushrooms are softened, and the leeks are softened.
Pour in the wine and cook for an additional 2 minutes. Introduce the cream or half and half, increase the heat to medium-high, and cook for 5 minutes until the cream coats the back of a spoon and slightly reduces. Stir in the last tablespoon of butter for added gloss, and adjust the seasoning to taste.
Add the chicken breasts back to the pan and top with the sauce, or transfer to serving plates and spoon sauce over them then.
Serve with mashed potatoes, rice, or noodles.