Pat chicken cutlets dry with paper towels. Season both sides evenly with kosher salt, black pepper, Italian herbs, paprika, and garlic powder.
Heat olive oil and 1 tablespoon butter in a large 12-inch cast-iron skillet over medium-high heat until hot. Add chicken cutlets and sear for 4 to 5 minutes per side until deeply golden and cooked through (165u00b0F / 74u00b0C internal). Transfer chicken to a clean plate and tent loosely with foil.
In the same skillet, melt 2 tablespoons butter over medium-high heat. Add sliced cremini mushrooms in a single layer and cook undisturbed for 3 minutes until deep golden brown. Stir and cook for another 2 minutes until tender. Add minced garlic and sautu00e9 for 60 seconds until fragrant.
Pour in chicken broth, using a wooden spatula to scrape up all the delicious golden browned bits (fond) stuck to the bottom of the skillet. Let reduce by half (about 2 minutes).
Lower heat to medium-low. Pour in heavy cream and whisk gently to combine. Bring to a gentle simmer for 3 minutes until the sauce slightly thickens.
Stir in freshly grated Parmigiano-Reggiano cheese and chopped parsley until the cheese melts into a silky, velvety cream sauce.
Return seared chicken cutlets and any resting juices back into the skillet. Spoon the rich mushroom garlic cream sauce generously over the chicken. Simmer for 1 to 2 minutes until everything is piping hot. Garnish with fresh thyme and serve immediately!