Step 1: Pounding and Seasoning. Place the chicken breasts between two sheets of plastic wrap and pound to an even 1/2-inch thickness. Season both sides with salt, pepper, and dried herbs, then dredge lightly in flour.
Step 2: Searing the Poultry. In a large skillet, heat the oil and butter over medium-high heat. Sear the chicken for 5-6 minutes per side until golden brown and almost cooked through. Remove to a plate.
Step 3: Sautéing Aromatics. In the same skillet, add the minced garlic and shallot. Sauté for 1-2 minutes until translucent and fragrant, taking care not to burn the garlic.
Step 4: Deglazing and Reducing. Pour in the white wine, using a spatula to scrape up the browned bits from the pan. Simmer for 3-4 minutes until the liquid has reduced by half.
Step 5: Finishing the Sauce. Stir in the heavy cream and chicken broth. Bring to a simmer, then whisk in the Parmesan cheese until smooth. Return the chicken to the pan and simmer for 3-5 minutes until the sauce thickens and the chicken is cooked through.