Season the chicken with salt and pepper on both sides. Heat 2 tablespoons of oil in a heavy-bottom skillet over medium heat.
Add chicken breast and cook until both sides are browned, and the inside is cooked through, about 12-15 minutes, depending on the thickness of your chicken breasts.
Once the chicken is done cooking, transfer to a chopping board and rest for 5 minutes, then thickly slice it.
Wipe the pan with a paper towel, add remaining 1 tablespoon of olive oil, and heat until hot.
Add garlic and sauté until fragrant. Pour stock, Dijon mustard, 1 cup milk, salt, and pepper. Reduce the heat to low and let it simmer until the liquid starts to bubble.
Meanwhile, whisk 1/4 cup of milk with flour and cornstarch. Pour in the milk cornstarch mixture; cook whisking constantly until the sauce starts to slightly thicken.
Stir in vinegar and grated parmesan cheese, whisk until sauce is thick and creamy.
Add in the cooked chicken breast and cook until chicken is warm through, about 2 minutes more.
Transfer the creamy parmesan chicken to a serving dish. Sprinkle some more parmesan on top if desired.
Garnish with fresh parsley leaves and enjoy!