Pat the chicken pieces dry with paper towels and season generously with salt and pepper. Let the chicken sit in the seasoning while you prepare the remaining ingredients.
Preheat a large oven-safe skillet or Dutch oven over medium-high heat. Add avocado oil. Place the chicken skin-side down and cook until the skin is golden brown and crisp, about 4-5 minutes per side. Remove the chicken from the skillet and set aside.
Reduce the heat to medium and add the diced onions. Season with a pinch of salt. Sauté until softened and translucent, about 3-4 minutes. Add the garlic and cook until fragrant, about 30 seconds.
Stir in the chicken stock, scraping brown bits off the bottom of the pan. Add Dijon mustard, whole grain mustard, and maple syrup or honey. Whisk until combined. Season with salt and pepper to taste.
Add the quartered radishes to the skillet, spreading them evenly across the bottom.
Nestle the browned chicken pieces on top of the radishes, skin-side up.
Scatter the torn mustard greens over the chicken. Cover the skillet and let the greens wilt over medium low heat for about 20 minutes. The liquid should be simmering.
Uncover the skillet and gently push the wilted greens underneath the chicken and radishes. Pour in the heavy cream and stir lightly to combine with the braising liquid.
Preheat the oven to 275°F. Cover the skillet partially with a lid or foil and transfer to the oven. Bake for at least 2 hours (can be up to 5-6), or until the chicken is tender, falls off the bone and the flavors meld beautifully. If you want it done faster, you can also cook it on low heat on the stove top, simmering, partially covered, and let it simmer for 1 1/2-2 hours (or more) until the meat is white and tender. If you’re nervous about slow cooking chicken, make sure the internal temp of the meat is at least 165F to ensure it’s done.
Serve hot with the radishes, greens, and creamy mustard sauce spooned over. Pair with crusty bread to soak up the sauce.