Pat the chicken dry, cut each piece in half, and season both sides with salt and pepper.
Heat a large skillet over medium-high heat. Add 1 tablespoon each of butter and oil. Once the butter is melted and bubbling, add the chicken.
Cook 4–5 minutes per side, until the internal temperature reads 165°F for a fully cooked piece — a little under is fine here, since it finishes in the sauce later. Remove, cover with foil, and set aside. Cook 4–5 minutes per side, until the internal temperature reads 165°F for a fully cooked piece — a little under is fine here, since it finishes in the sauce later. Remove, cover with foil, and set aside. Add another tablespoon each of butter and oil to the same pan. When melted, add the mushrooms in a single layer — don't stir them right away. Let them sit over medium heat for 5 minutes, until browned on one side, then toss to finish. Remove and set aside. Add the remaining butter and oil (2 tablespoons each) to the pan. Sauté the garlic over medium-low for about 2 minutes, until just light brown — watch it closely, since it goes from golden to bitter fast.
Sprinkle in the flour and whisk constantly for a few seconds, just until it disappears into the butter and garlic — that's the roux. Drizzle in the chicken broth and half & half, whisking as you go, and bring to a simmer. Once simmering, stir in the grated cheese until it melts into the sauce, then season with salt and pepper. The sauce is ready when it coats the back of a spoon and holds a clean line if you run a finger through it.
Add the spinach and stir until it wilts into the sauce — this takes less than a minute.
Return the mushrooms to the pan, then nestle the chicken pieces into the sauce along with any juices they released. Simmer another 3 minutes over medium-low heat until the chicken is fully cooked through. Plate the chicken and spoon the sauce over everything.