Cut your chicken breasts in half lengthwise to make four thinner cutlets. Season them with salt & pepper.
Add the olive oil to a skillet over medium-high heat.
Once the pan is hot, add the chicken. Cook it for 4-5 minutes/side until golden. Transfer the chicken to a plate.
Add the butter to the skillet. Let it melt, then add in the mushrooms. Cook the mushrooms, stirring occasionally, until the water has been released, it cooks off, and they get a good sear.
Add the garlic, Italian seasoning, white wine, and balsamic vinegar to the skillet. Let it bubble until the liquid is reduced by half.
In a small bowl, mix the cornstarch with 2 teaspoons of cold water and stir until dissolved.
Reduce the heat to medium. Add the cream and cornstarch slurry to the pan and give it a good stir.
Add the chicken back to the skillet. Simmer for another 5 minutes or so until the chicken is cooked through and the sauce has thickened a bit. Season with salt & pepper as needed and sprinkle with parsley if using.