Pat the chicken dry, cut each piece in half,and sprinkle salt and pepper on both sides.
Heat a large frying pan over medium high heat. When heated, add 1 tablespoon each of butter and oil.
When the butter is melted and bubbling, add the chicken pieces. Cook on each side for4-5 minutes, until internal temperature reaches 165° F.
Remove the chicken from the pan,cover with foil and set aside.
Add another tablespoon of each butter and oil to the pan. When the butter is melted, add the mushrooms in a single layer. Cook for 5 minutes over medium heat, until browned. Remove the mushrooms from the pan and set aside.
Add the remaining butter and oil(2 tablespoons of each) to the pan. When heated,sauté the garlic for about 2 minutes, until light brown. Then sprinkle on the tablespoon of flour, whisking to blend as a roux. This will only take a few seconds. Drizzle in the chicken broth and half & half and bring to a simmer.
Once the sauce is simmering, add in ½ cup grated cheese and stir to blend. Add in salt and pepper to taste.
When blended, add the 2 cups of spinach and stir until the spinach is wilted into the sauce.
Return the mushrooms to the pan and stir. Nestle the chicken pieces into the sauce in the pan, along with the juices from the chicken, and cook for about 3 more minutes over medium low heat.
Plate the chicken and smother with sauce.