Pat chicken cutlets dry with paper towels. Season both sides generously with kosher salt and cracked black pepper. Dredge lightly in flour, shaking off any excess.
Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large deep skillet (cast iron preferred) over medium-high heat until sizzling. Add chicken cutlets in a single layer and sear for 4 to 5 minutes per side until golden brown and cooked to an internal temperature of 165°F (74°C). Transfer chicken to a warm plate and tent with foil.
In the same skillet, add the remaining 1 tablespoon of butter and 1 tablespoon of olive oil. Add sliced cremini mushrooms and sear undisturbed for 3 minutes until caramelized and browned. Stir in the diced shallot, minced garlic, and fresh thyme; sauté for 1 minute until fragrant.
Pour in the chicken broth, using a wooden spoon to scrape up all the flavorful caramelized browned bits (fond) stuck to the bottom of the skillet. Simmer for 2 minutes until liquid reduces by half.
Reduce heat to medium-low and pour in the heavy cream. Bring to a gentle simmer for 3 minutes until the sauce thickens into a glossy, velvety consistency.
Stir in the freshly grated Parmigiano-Reggiano until melted and completely smooth. Taste and adjust salt and pepper as needed.
Return the seared chicken cutlets and any resting juices back to the skillet. Spoon the bubbling garlic parmesan mushroom sauce generously over the chicken. Simmer for 1 to 2 minutes to reheat through.
Garnish with freshly chopped Italian parsley and extra grated parmesan. Serve immediately while piping hot.