Transfer one chicken breast to a large ziplock bag and seal. Using a rolling pin or meat tenderizer, pound your chicken until it is approximately 1/2-3/4 inch thickness. Set aside to a clean plate and repeat with the remaining chicken breasts. Pat the chicken dry with paper towels and generously season both sides with kosher salt and black pepper.
Add the all-purpose flour to a (new) large zip-lock bag. Use kitchen tongs to transfer 3 chicken breasts to the zip-lock bag filled with flour and seal. Gently toss to coat the chicken in flour.
Add one tablespoon of butter and one tablespoon of olive oil to a large 12-inch skillet set over medium-high heat. Heat until melted and shimmering. Transfer the dredged chicken breasts to the skillet, gently shaking off any excess flour before adding to the skillet. Cook until golden brown, approximately 5-7 minutes on each side. Remove the chicken to clean plate and cover with foil to keep warm. Repeat with the remaining chicken.
Over medium heat, melt 2 tablespoons of butter in the same skillet you used to cook the chicken. Add the chopped onions and cook, stirring frequently, for 3-4 minutes, or until soft. Add the garlic and continue to cook for an additional minute before adding the wine. Increase heat to high, scrape up any browned bits stuck to the bottom of the pan, and allow the liquid to reduce by at least half.
Reduce heat back to medium-high, and stir in the sliced mushrooms. Continue to cook until the mushrooms start to soften, approximately 3-4 minutes.
Sprinkle the mushrooms and onions with 2 tablespoons flour and mix well combine. Slowly stir in the chicken broth, dissolving any clumps that may have formed with the flour.
Add the spinach, working in batches if needed. Cook until wilted, stirring as needed.
Stir in the whole milk and heavy whipping cream, mixing well to combine with the spinach, mushrooms, and onions. Bring your sauce to a gentle boil and stir in the grated parmesan cheese. Reduce heat to medium.
Gently nestle the cooked chicken back into the pan and turn to coat in the sauce. Allow the chicken to simmer for approx 4-5 minutes, or until heated through.
Serve immediately with cooked rice, pasta, or mashed potatoes, if desired. Enjoy.