In a large bowl, mix breadcrumbs and cheese together. In a separate bowl whisk together, the eggs and a little salt and pepper. Take chicken breasts and cut into 1-inch strips. If you want bigger portions then cut accordingly. Now take each piece of chicken and dip into egg wash and then into breadcrumb/cheese mixture. To insure a good coating, push down on the chicken as you are rolling it in the breadcrumbs. As each piece is breaded, place on a plate. When you are finished, wrap plate with plastic wrap and put in the fridge for at least 30 minutes. You can do this step several hours ahead of time.
In a large heavy skillet, heat the butter and oil, when butter foams, add the chicken and cook over medium heat for 3-4 minutes on each side, or until it has a nice golden color. Now add the vermouth to the chicken, put your lid on and lower the heat. You want to let it simmer for about 10-12 minutes. Now you should have only a few tablespoons of sauce left in the skillet. At this point, you need to transfer the chicken to a plate. I like to put the chicken in the microwave to stay warm. You could also turn you oven on just long enough to get it warm and place chicken in there.
Raise your heat and add a few tablespoons of olive oil. Once the oil has heated up add mushrooms and cracked pepper. The mushrooms will look like a lot, but they will shrink. Cook for about 8-10 minutes stirring frequently. Now add your cream to the pan. You do not want the heat to high as this will burn the cream and ruin it. Once the cream is boiling, begin stirring it around until it starts to get thick, this should take about 10 minutes or so. If you want a thicker sauce you need to simmer longer. Now is the part were salt is added, this is what brings the sauce together. When adding salt, do so a little at a time. You can always add more, but you cannot take away. Start out with a 1/2 teaspoon of salt, and see if that is to you liking, add more if needed. You will know your there when the sauce begins to not taste so much like straight cream. As far as the pepper, which is completely up to you, add as much as you like. I like a lot of fresh pepper in mine, but that’s me. Assuming this process did not take to long, your chicken should still be nice and warm. If not you can give it a quick 1 minute zap in the microwave. If you used the oven method, be careful the plate will be hot. Now take the chicken, put on individual plates and spoon sauce over, and enjoy. This dish can be served over noodles or even rice. Just be sure to start either with enough time to finish when everything else is done. I hope you enjoy!