Pat the chicken breast cutlets dry with a paper towel and season with a pinch of salt and pepper on each side. If chicken breast is not sliced thinly already, simply cut lengthwise (butterflying) the chicken breast, and cut all the way through to result in two thin chicken breasts before seasoning.
Dip each chicken breast cutlet in the flour and shake off excess. Set aside.
Heat a large skillet over medium heat for a few minutes. Add a few tablespoons of cooking oil and continue to heat for 30 seconds.
Place the chicken in the skillet, being sure to not overcrowd the skillet, and sear until browned on each side, about 3-4 minutes. Remove chicken and set aside.
Continue heating the skillet on medium and add more cooking oil if needed. Add the sliced mushrooms and stir to coat.
Sauté the mushrooms until browned. At the end of cooking time, add a pinch of salt & the fresh thyme.
Turn the skillet to medium-low, and add the 4 tbsp of butter to melt.
Once melted, add the minced garlic and sauté until fragrant, about 30 seconds.
Pour in the half & half, and bring to a gentle simmer. Simmer for a few minutes until thickened, stirring frequently.
Add the grated Parmesan cheese and and a pinch of pepper. Stir until cheese is melted. Taste to see if salt is needed for preference.
Add the chicken back to the skillet and simmer on low for another few minutes. Garnish with extra thyme.