Combine all the marinade ingredients – yogurt, lime juices, and cayenne pepper - in a good-sized mixing bowl. Lightly prick the skin of the chicken thighs with the tip of a sharp knife. This allows the marinade to penetrate more easily and also means the skin won’t contract too much when the thighs get their turn under the hot broiler. Add the chicken to marinade and use your fingers to make sure the thighs get thoroughly coated. Set aside for 40 minutes.
Braising the chicken thighs
Set a big skillet (12-inch is grand) on a high heat and add the butter and olive oil. As soon as the buttery oil starts to foam, add the thighs, skin side up. Let the chicken sizzle hotly for 3 minutes, and then drop the heat to medium-low. Do not turn the chicken, but just let it cook on that medium-low heat for another 5 minutes. You’re looking to get a deep golden colour only on the underside of the thighs.
Now cover the skillet with a tight-fitting ‘lid’ of silver foil and drop the heat to its lowest setting. Allow to cook like this for 10 minutes. Then turn off the heat and let the chicken sit in the skillet under the tight ‘lid’. What you’re aiming for here is to poach the chicken in skillet’s steamy juices.
For the potatoes
Set your oven to 350 degrees Fahrenheit. Add the olive oil to a heavy metal baking dish big enough to hold all the potatoes in a single layer, and set the dish in the oven while it's heating.
Put the potatoes in a saucepan with a level teaspoon of ground sea salt and just enough cold water to cover them. Set the pan on a high heat and bring to the boil. Turn the heat to low, cover the pan, and allow the potatoes to simmer for 5 minutes. Turn off the heat, drain the potatoes and return them to their pan. Cover the pan and let the potatoes steam there for 5 minutes.
Remove the baking dish from the oven, and set it onto a medium heat on the stovetop. Quickly add the potatoes, salt, black pepper, and garlic cloves. Stir so the potatoes get well coated in the salt, pepper, and sizzling oil. Remove from the heat, add the rosemary sprigs, and cover the dish with a sealing sheet of tin foil. Return the dish to the oven on a high shelf and increase the heat to 425 degrees Fahrenheit. Bake for 15 minutes. (That tight foil covering matters because it ensures the potatoes become infused with wonderful flavors of garlic and rosemary.)
After that first 15 minutes in the oven, remove the dish and give all the potatoes a quick turn. Return them to the oven – uncovered - to roast for another 10 mins. Done.
For the sauce
While the potatoes are roasting away in the oven, you can make the sauce. Start by adding the olive oil and butter to a saucepan set on medium heat. Once the butter melts, add the onions, garlic, chilis and salt. (Adding the salt now helps pull out their flavors.) Stir well. Drop the heat to low-medium and cook for 10 minutes, with the occasional stir until the onions have really softened – but not taken on any color. Add the cherry tomatoes, bay leaves and sugar, stir, and cook for another 15 minutes on that low-medium heat. Break up the tomatoes a little as they soften with some bold stirring but do let them keep some of their body.
Tip in the sliced peppers and tomato puree. Stir and raise the heat to medium-high until the sauce just bubbles. Drop the heat to low and cover the pan. Cook for 15 minutes, stirring a few times - there’s not yet much liquid yet in the sauce.
Stir in half stock, and let the sauce cook covered for 10 minutes. Now add the remaining stock, olives and capers. Stir. Let in cook for another 5 minutes on low heat just so that the peppers are al-dente. Check for salt and chili heat, adjust to taste. Done.
Bringing it all together
Set your broiler to medium-high. Pour the sauce into a good-sized oven-proof serving dish. Arrange the chicken carefully on top of sauce – just so it lays there, don’t push it down into the sauce. Baste the chicken with the pan juices and set the dish under the broiler until the chicken skin is nicely browned and crisp. Serve.
Plate the chicken with sauce and potatoes alongside.
To drink? I’d keep things strongly Mediterranean and go for something light and fruity from, say, Italy. And for me, with a dish like this, that means a Chianti.