Thinly slice the baby Bella mushrooms and shallot. Set aside.
Tear the mairake mushrooms with your hands into small pieces. No need to slice. Set aside.
Thinly slice the garlic.
Using a rolling pin or meat mallet, place chicken in a resealable bag and pound out until evenly ½” thick.
Season chicken with salt and pepper, and dredge with the flour on both sides.
In a large cast iron skillet or deep frying pan, heat butter and oil over medium heat until melted and bubbly about 3-4 minutes.
Add floured and seasoned chicken to the skillet, and cook 5-6 minutes a side, until just browned.
Transfer chicken to a rimmed plate and set aside.
Add the shallots. And mushrooms to the pan, and season with salt and pepper.
Cook, stirring occasionally 6-8 minutes over medium heat until mushrooms are softened.
Add rosemary, thyme (do not chop herbs) and garlic, and stir to coat. Cook 1 minute until just fragrant.
Add chicken stock, milk and parmesan and cook on medium-high heat for 4-5 minutes, stirring occasionally until slightly thickened.
Add chicken (and accumulated juices) back to the pan, reduce heat to medium and cook 8-10 minutes, until sauce is slightly thickened.
Slice chicken and serve with mashed potatoes, polenta, or rice.