Slice chicken breasts in half horizontally to create 4 even cutlets. Mix flour, salt, and pepper in a shallow dish, and lightly dredge the chicken, shaking off excess.

Heat olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Sear chicken for 4-5 minutes per side until golden brown and cooked through. Remove to a plate.

Reduce heat to medium, add 1 tablespoon of butter and the sliced mushrooms. Cook undisturbed for 5 minutes to brown, then stir and cook 2 more minutes until caramelized.

Add chopped shallot and cook for 1 minute. Add minced garlic and fresh thyme, cooking for 30 seconds until fragrant.

Pour in the white wine, scraping the bottom of the skillet to release browned bits. Let simmer until reduced by half.

Stir in the chicken broth and simmer for 3 minutes. Reduce heat to medium-low, slowly stir in the heavy cream, and simmer until the sauce thickens and coats a spoon.

Stir in the final 1 tablespoon of butter for gloss. Return the chicken to the skillet to warm through for 1 minute. Garnish with fresh parsley and serve immediately.