Slice chicken breasts in half horizontally to make thin cutlets. Pat dry with paper towels. In a shallow bowl, whisk flour, garlic powder, smoked paprika, salt, and black pepper. Dredge cutlets in seasoned flour, shaking off excess.

Heat olive oil and 2 tbsp butter in a large skillet over medium-high heat. Arrange 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 clean platter.

Reduce heat to medium. Add remaining 2 tbsp butter and sliced cremini mushrooms to the skillet. Sauté undisturbed for 2-3 minutes until browned, then stir. Add minced shallots and garlic; sauté for 2 minutes until fragrant and soft.

Pour chicken broth and fresh lemon juice into the pan. Scrape up all caramelized browned bits from the bottom with a wooden spoon. Simmer for 3 minutes until liquid reduces by half. Turn heat down to low and stir in heavy cream, lemon zest, and grated Parmesan. Simmer gently for 2 to 3 minutes until sauce thickens into a silky coating.

Return seared chicken cutlets and accumulated juices to the pan. Spoon creamy mushroom sauce generously over the chicken to warm through for 1 minute. Transfer to a white platter, pour remaining sauce over top, and scatter chopped fresh parsley before serving.