Pat chicken legs and thighs dry. Season generously with salt and pepper.

Heat olive oil in the Dutch Oven over medium-high heat. Sear chicken skin side down until deeply golden, about 6–7 minutes. You'll know it's ready to flip once it releases from the pan. It sticks because the crust is still forming, so don't nudge it around if it's not releasing. The protein will almost always tell you when it's ready. Flip and sear 3–4 minutes more. Remove and set aside.

Reduce heat to medium. Add leeks and a pinch of salt. Cook, stirring, until softened and caramelized, 6–8 minutes.

Add garlic and cook until fragrant, 1 minute. Deglaze with white wine, scraping browned bits from the pan. You'll know all of the bits have released when you run your spoon over the bottom of the pan and it feels smooth. Raw wine throws a sharp scent that pushes your face back. Step away a few seconds, then come in from the side. You'll know it's ready when that scent is gone and you smell the wine's own fruit and the leek instead.

Return chicken to the Dutch Oven, skin side up. Add chicken stock, lemon juice, lemon slices, lemon peel, and thyme. Liquid should come about halfway up the chicken.

Bring to a simmer, then cover and transfer to a 350°F oven. Braise for 45-60 minutes (until chicken is tender and reaches 175°F internally).

Remove chicken to a plate and tent with foil. Discard thyme sprigs and lemon peel.

Reduce sauce on the stovetop over medium heat until slightly thickened, 5–7 minutes.

Off heat, whisk in butter and crème fraîche. Stir in the capers, then return the chicken to coat lightly.

Heat olive oil in the Cast Iron Skillet over medium heat. Add onions and cook until softened, 4 minutes.

Add mushrooms, thyme, salt, and black pepper to taste. Sauté until golden and tender, 5–6 minutes.

Season to taste with kosher salt.

Put a handful of arugula on each plate and spoon the chicken, mushrooms and sauce over it. The leaves wilt into the sauce, which is the reason they go underneath rather than alongside.