Season the chicken on both sides with salt and pepper, then lightly dredge in flour, shaking off excess.
Heat 2 tablespoons olive oil and 1 tablespoon butter in a large skillet over medium-high heat.
Add the chicken in a single layer and cook for 3–4 minutes per side until golden brown. Remove and set aside.
In the same pan, add 2 tablespoons olive oil and 1 tablespoon butter, then add the mushrooms.
Cook the mushrooms for 5–8 minutes until softened and lightly browned, seasoning lightly with salt and pepper. Remove and set aside with the chicken.
Add the remaining 1 tablespoon olive oil to the pan, then sauté shallots and garlic for about 1 minute until fragrant.
Sprinkle in the remaining 1 tablespoon flour and stir for 1 minute to cook off the raw flour taste.
Pour in the beef stock and water, scraping up any browned bits from the bottom of the pan. Let the sauce simmer for about 2 minutes until slightly thickened.
Return the chicken and mushrooms to the skillet, including any juices from the plate.
Add the remaining tablespoon of butter and thyme, then let everything simmer together for 2–3 minutes until the chicken is fully cooked and coated in the sauce.
Spoon the sauce over the chicken, garnish with parsley, and serve warm.