Place each chicken breast between sheets of plastic wrap and pound to an even thickness of about 2 cm; season generously with salt and pepper.
Heat a tablespoon of oil and half a tablespoon of butter in a heavy skillet over medium-high heat until shimmering and fragrant.
Place the chicken breasts in the skillet and sear for about 3-4 minutes on each side until golden brown and cooked through, then transfer to a plate to rest.
Add sliced mushrooms to the same skillet and sauté over medium heat, stirring occasionally, until deeply browned and smoky, about 5-7 minutes.
Add minced garlic to the mushrooms and cook for about 30 seconds until fragrant, the kitchen filling with earthy aroma.
Pour in the heavy cream and bring to a gentle simmer, stirring constantly to prevent sticking, until the sauce thickens slightly, about 2 minutes.
Return the chicken breasts to the skillet, spoon sauce over the top, and simmer on low heat for about 5 minutes to meld the flavors and heat through.
Stir in lemon juice and chopped thyme, taste for seasoning, and adjust with more salt or pepper if needed.
Remove the chicken breasts and let rest for 3 minutes, then slice and serve generously topped with the velvety mushroom sauce.
