Place each chicken breast on a cutting board. Place 1 palm on top of 1 breast to hold it still; using a chef’s knife with other hand, slice breast in half horizontally. Repeat process with second chicken breast. Place chicken breast halves between 2 sheets of parchment paper or plastic wrap. Using a meat mallet or small skillet, pound each half into an even thickness. Sprinkle both sides evenly with salt and ½ teaspoon of the pepper.

Heat 1 tablespoon each of the oil and butter in a large skillet over medium-high. While butter mixture heats, place flour in a shallow dish; dredge 2 seasoned chicken halves in flour, and shake off excess. Add dredged chicken halves to hot butter-oil mixture; cook until browned on bottom, 3 to 4 minutes per side. (Adjust heat as needed so oil vigorously, but not violently, bubbles around chicken.) Transfer cutlets to a large plate. Repeat cooking process with 1 tablespoon each of the butter and oil and chicken halves. Wipe out skillet.

Heat remaining 2 tablespoons oil in skillet over medium-high. Add mushrooms and remaining ¼ teaspoon pepper; stir to coat mushrooms in oil. Cook, stirring occasionally, until mushrooms are lightly browned, about 8 minutes. Add shallot, garlic, and thyme; cook over medium, stirring often, until fragrant and shallot softens slightly, about 1 minute.

Stir wine and sherry into skillet; bring to a simmer over medium-high, and simmer, undisturbed, until alcohol smell dissipates, about 2 minutes. Add stock; bring to a simmer over medium-high; simmer, stirring occasionally, until slightly reduced, about 3 minutes.

Add heavy cream to skillet. Bring to a simmer over medium-high; simmer, stirring occasionally, until just slightly reduced, about 2 minutes. Add spinach in 2 batches, stirring after each addition until all leaves are wilted and incorporated into sauce. Nestle chicken in sauce, and cook until heated through, about 1 minute. Remove from heat. Slowly stir in Parmesan until completely melted. Serve immediately.