Combine the minced garlic, shallot, lemon zest, lemon juice, mustard, paprika, cayenne pepper, oregano, and thyme sprigs in a bowl to create the marinade base. Add the olive oil and water, then stir until well combined. Pat the chicken pieces dry and coat them thoroughly with the marinade, ensuring all surfaces are covered. Transfer the marinated chicken to a container, cover, and refrigerate for at least 12 hours, though 24-48 hours will develop even deeper flavor.

Preheat your oven to 425°F. While the oven heats, cut the potatoes into 1.5-inch wedges and set aside.

Heat 3 tablespoons of olive oil in a large oven-safe skillet over high heat until it's shimmering and almost smoking. Add the potato wedges in a single layer and sear them for 3-4 minutes per side until they develop a golden-brown crust.

Remove the marinated chicken from the refrigerator and arrange it in a large roasting pan or oven-safe skillet along with all its marinade. Season the potatoes with salt, freshly ground black pepper, 2 teaspoons of oregano, and minced garlic. Arrange the potatoes around the chicken in the same pan, cover with foil, and place on the top rack at 425°F for 45 minutes.

Remove the foil and continue baking for 30-35 minutes until the chicken skin is golden brown and crispy. The potatoes should bake on the lower rack for approximately 20 minutes total, tossing them halfway through.

Once the chicken is beautifully browned and cooked through, carefully remove it from the roasting pan and set aside. Pour in 1 cup of chicken stock to the cooking liquid and simmer for 4-5 minutes, scraping the bottom to release any flavorful browned bits. Whisk in the cold butter at the very end.

Pour the sauce over the chicken and potatoes just before serving.