Slice each chicken breast in half lengthwise and pound thin with the meat mallet.
Season each chicken cutlet generously with salt and pepper.
Top each cutlet with one slice of ham follwed by 1/4 cup of Swiss cheese.
Roll each chicken breast from one end to the other and secure with a toothpick.
Heat the avocado oil in a large skillet over medium high heat.
Brown each chicken breast on all sides until the internal meat thermometer reads 165F.
Remove the chicken from the pan.
Reduce the heat of the skillet to medium and add the mushrooms. Sauté until the mushrooms are soft, about five minutes. Add in the garlic and saute' 1 minute more.
Add in the heavy cream and let simmer until thickened, about 5 minutes.
Remove the tootpicks from the chicken and slice into rounds. Serve with mushroom sauce on top and chopped parsley (optional).