Start by seasoning the chicken thighs generously with salt. Heat oil in a large skillet over medium-high heat, then add the chicken skin-side down. Allow the chicken to sear for about 5 minutes, or until the skin turns golden and crispy. Flip and cook the other side for another 2 minutes before removing the chicken from the pan.
Leave about 2 tablespoons of rendered fat in the skillet. Add mushrooms and sauté until they develop a brown, caramelized color. Then, stir in the onions, garlic, and shallots. Cook for 5 minutes, letting the onions soften and release their sweetness.
Sprinkle flour over the vegetables, stirring until fully incorporated. Slowly pour in white wine, scraping up any browned bits from the pan to maximize flavor. Bring to a boil and simmer for 3-5 minutes until the liquid reduces by half. Then, add chicken broth and thyme.
Return the chicken thighs (and any juices) to the pan, placing them skin-side up. Cover with a lid and simmer on low heat for 45 minutes, turning the chicken halfway through cooking. Remove the lid and simmer uncovered for an additional 10-15 minutes while the sauce thickens further.
Once the chicken is fork-tender, stir in the heavy cream. Raise the heat slightly to help the cream integrate into the sauce. You can reduce the sauce further to your desired consistency. Garnish with fresh thyme sprigs before serving.