For the lobster stock: In a large saucepan, heat the olive oil over medium-high heat and add the lobster bodies. Stir with a wooden spoon for 5–7 minutes, until the lobsters give off an intense aroma. Add the tomato paste and stir to thoroughly coat the lobster bodies. Deglaze the pan with the brandy and cook off the alcohol, about 1 minute. Add the onions, carrots, tomato, fennel, garlic, and basil and sauté for 3–5 minutes, until the vegetables begin to take on color. Add 6 cups of the reserved lobster water, reduce the heat to medium low, and simmer for 2–3 hours.
For the lobster stock: Strain the stock or if you want more flavor run the whole darn thing through a food mill; this will extract all the delicious juices from the lobster bodies and give you a rich stock. Remove about a cup of the stock and steep a small pinch of saffron in the liquid for 3 or 4 minutes, then return the saffron-infused broth to the stockpot. (Steeping saffron in a bit of liquid maximizes the spice’s flavor and color.)
For the sofrito: Bring 2 cups water to a boil in a small saucepan. Remove from the heat, add the
dried peppers, and soak until soft, about 10 minutes. With a small knife, cut the peppers in half and remove and discard the seeds. Carefully scrape the flesh away from the skin and discard the skins. Set aside the flesh.
For the sofrito: In a large heavy-bottomed pot, heat the olive oil over medium-high heat. Add the onions and sweat until translucent, about 5 minutes. Add the flesh from the dried peppers, bell peppers, grated tomatoes, garlic, vinegar, and salt and pepper. Lower the heat and simmer, stirring periodically, for at least 2 hours. It takes that long to develop the sugars and bring out the sweetness of the vegetables, which is so important to the characteristic rich flavor of sofrito.
For the paella: Cut each lobster tail crosswise into 3 pieces with a sharp knife, leaving the shell on. Set the tails and claws aside. In a large paella pan over an outdoor grill or on a stovetop, heat the olive oil over medium/high heat. If you’re doing the paella on the stovetop, you can use two burners if your pan is large.
Season the chicken parts and shrimp with salt and pepper and sear in the paella pan. Once they’ve turned nicely golden, remove the shrimp and set aside. Add the chorizo to the pan with the chicken and cook to slightly render the oil, about 2 minutes.
Add the sofrito and about two-thirds of the stock and bring to a rolling boil. Add all the rice and gently stir until evenly mixed. It’s important that the stock be boiling when you add the rice so the grains release their starch and create the slightly creamy texture that binds the grains of rice together.
Season with salt and move to the cooler side of the grill or reduce the temperature to medium. You want the stock to be gently bubbling over the rice. Throughout the remainder of the preparation, add the remaining hot stock little by little as the rice absorbs the liquid until you’ve used all the stock. After the rice has simmered for 10 minutes, bury the mussels and clams in the rice. Cook for 4 minutes, then evenly distribute the lobster tails and claws and the shrimp and cook for another 4 minutes.
At this point the rice should have absorbed all of the stock and the grains should be plump and moist with a slight tooth but no chalky starch. Discard any unopened mussels or clams. Remove the paella from the heat and let rest for 5 minutes before serving.