Use a tenderizer to pound the chicken on both sides. Really use some elbow grease here!
In a large skillet over medium heat, pour in 1 tablespoon of the olive oil. Add the chicken thighs or breasts, season with salt and pepper, and cook until golden brown on both sides and no longer pink in the middle, about 4 to 6 minutes per side (depending on the thickness of the chicken). Transfer to a clean plate.
Wipe out the skillet and add remaining 2 tablespoons of olive oil. Still on medium heat, cook the shallots and mushrooms, while stirring, until they soften and the mushrooms darken in color, 4 to 5 minutes.
Add the garlic and thyme and cook, while stirring, for another minute.
Pour in the white wine, turn up to medium-high heat, and cook for a couple minutes until reduced down almost all the way. Stir in the chicken broth, cream, salt, and pepper.
Add the chicken back into the pan and cook until sauce comes together and the chicken is reheated all the way through. Sprinkle with more fresh thyme for garnish (optional) and serve over polenta (recipe below) if desired.