Slice chicken breasts in half lengthwise to create 4 thinner cutlets, then pound to 1/2-inch thickness if needed. Pat completely dry with paper towels.
In a shallow bowl, mix flour, garlic powder, salt, and pepper. Press each cutlet into the flour mixture, coating both sides, and shake off excess.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add chicken in a single layer and cook 4-5 minutes per side until golden brown and 165°F internally. Transfer to a plate.
Reduce heat to medium, add 1 tablespoon butter and minced shallot. Cook 2 minutes until softened, then add garlic and cook 30-60 seconds until fragrant.
Pour in white wine and scrape up browned bits from the pan. Add chicken broth and Italian seasoning, bring to a simmer, and cook 5-7 minutes until reduced by half.
Pour in heavy cream and bring to a gentle simmer over medium heat. Cook 2-3 minutes until slightly thickened, then stir in remaining 1 tablespoon butter until melted and incorporated.
Add fresh spinach and stir until wilted, about 1-2 minutes. Sprinkle in Parmesan cheese and stir until melted. Taste and adjust seasoning with salt and pepper.
Return chicken to the skillet, spooning sauce over the top. Simmer gently for 2-3 minutes to heat through. Serve immediately over pasta, rice, or mashed potatoes with extra Parmesan and red pepper flakes if desired.