Prep the Chicken: Pat the chicken thighs completely dry with paper towels; this is key for getting crispy skin. Season generously on both sides with salt and pepper. Heat 1 tablespoon of olive oil in a large skillet over medium-high heat until shimmering.
Sear the Chicken: Place the chicken in the hot pan, skin-side down. Let it cook undisturbed for about 6 to 7 minutes until the skin is deep golden brown and crispy. Flip carefully, season again with a pinch of salt, and cook for another 5-6 minutes until the internal temperature reaches 160°F (71°C). Remove chicken from the pan and set aside on a plate; do not cover yet to let the skin stay crisp.
Cook the Aromatics: Reduce heat to medium. Add the remaining 1 tablespoon of oil to the same pan. Toss in the sliced mushrooms and cook, stirring occasionally, for about 5-7 minutes until they release their water and turn a deep brown color (don't rush this step!). Push the mushrooms to the side or remove them temporarily if you want to sauté the garlic separately to prevent burning.
Build the Sauce: Add the minced garlic to the pan (either with the mushrooms or in the empty space) and cook for just 30 seconds until fragrant. Pour in the white wine or broth, scraping up any browned bits stuck to the bottom of the pan—that's where the flavor lives! Let it simmer until reduced by half.
Simmer and Finish: Stir in the heavy cream, Parmesan cheese, thyme, and red pepper flakes. Bring to a gentle bubble and cook for 2-3 minutes until the sauce thickens slightly and coats the back of a spoon. If using, stir in the mushrooms now if they weren't cooked separately.
Add Greens and Rest: Toss in the spinach or kale and stir until it wilts completely, about 1 minute. Return the seared chicken thighs to the pan, nestling them into the sauce so they are mostly covered. Spoon any extra sauce over the top. Let everything rest together for 5 minutes before serving to let the flavors meld.
