Pat the chicken thighs dry and season them generously with salt, pepper, and a light pinch of oregano. A dry surface helps achieve crisp, flavorful skin during searing.
Heat olive oil in a large, heavy skillet over medium-high heat. Sear the chicken thighs, skin-side down, until the skin is golden brown and crisp, about 6-8 minutes. Flip and sear the other side for 2-3 minutes. Remove to a plate and set aside.
In the same skillet, add the sausage slices (or links cut into chunks) and cook until browned and any fat renders, about 4-5 minutes. Transfer to the plate with the chicken, leaving the rendered fat in the pan.
Add the chopped onion to the skillet and sauté until translucent and starting to brown, about 4 minutes. Mince the garlic and add it, cooking for about 30 seconds until fragrant.
Stir in the bell pepper and cook for 2-3 minutes, until slightly softened. Deglaze the pan with the white wine, scraping up any browned bits from the bottom. Let the wine reduce by half, about 2-3 minutes.
Return the chicken and sausage to the pan. Add the chicken stock, lemon zest, and lemon juice. If using cherry peppers or giardiniera, add them now for extra kick and tang.
Bring to a gentle simmer, then reduce the heat to medium-low. Cover and braise for 15-20 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) and the flavors meld.
Uncover and let the sauce simmer a few more minutes to reduce slightly. Swirl in the butter for a silky finish and season to taste with salt and pepper.
Serve hot, spooning the peppers and pan sauce over the chicken. A side of crusty bread or creamy mashed potatoes pairs beautifully to soak up every bit of goodness.