Pat the chicken thighs dry with paper towels. Season both sides with kosher salt and black pepper.
Heat olive oil in a large skillet over medium-high heat. Add the chicken thighs skin-side down and cook until the skin is golden brown and crispy, about 6-7 minutes. Flip and cook the other side for 5 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, add the sliced mushrooms and chopped onion. Cook, stirring occasionally, until the mushrooms are browned and the onion is translucent, about 6 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom. Let it simmer until the wine is reduced by half, about 3 minutes.
Add the chicken broth, heavy cream, and fresh thyme leaves. Stir to combine and bring the mixture to a gentle simmer.
Return the chicken thighs to the skillet, skin-side up. Reduce the heat to medium-low, cover, and cook until the chicken is cooked through and the sauce has thickened slightly, about 12 minutes.
Remove the chicken thighs again and stir the Parmesan cheese and butter into the sauce until melted and smooth.
Return the chicken to the skillet to coat with the sauce. Sprinkle with chopped fresh parsley before serving.