Place a large skillet over medium heat. Melt 1 tablespoon butter and olive oil together. Cook chicken for 5 minutes per side until golden brown and no longer pink.
Remove chicken from pan, transfer to a plate, and cover to keep warm.
To the same pan, add mushrooms and a pinch of salt and saute until tender. Remove from pan. Wipe out the pan.
Return skillet to heat. Melt remaining butter and garlic. Cook for 30 seconds to 1 minute until fragrant. Stir in flour and cook for 1 minute to remove the raw flour taste.
Add wine to the skillet to deglaze and simmer for 3 minutes until the sauce begins to thicken.
Add milk and cook, stirring frequently, until sauce thickens. Add Parmesan cheese. Continue to stir until cheese is melted. Add salt and pepper to taste.
Add spinach to the sauce and cook until wilted Add mushrooms back in and stir into sauce.
Return chicken to pan. Spoon sauce over chicken. Serve warm.