Pat the chicken thighs dry with paper towels and season generously on both sides with kosher salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken thighs, skin-side down, and cook for 8-10 minutes until the skin is golden and crispy. Flip and cook for another 5 minutes, then transfer to a plate.
In the same skillet, reduce heat to medium. Discard all but 1 tablespoon of chicken fat, then melt the butter. Add the minced garlic and shallot, sautéing for 2 minutes until fragrant and softened.
Pour in the dry white wine to deglaze the pan, scraping up any browned bits from the bottom. Simmer for 3 minutes until the liquid reduces by half.
Stir in the chicken broth, heavy cream, and fresh thyme. Bring the sauce to a gentle simmer.
Return the chicken thighs to the skillet, skin-side up. Let simmer for 10-15 minutes, or until the chicken is cooked through (internal temperature reaches 165°F).
Remove from heat, stir in the grated Parmesan cheese until melted into the sauce, and garnish with fresh chopped parsley before serving.