(Note: Lobster Stock recipe not shown in video. Make Lobster Stock before commencing with the rest of the paella, or substitute chicken stock.) Lobster Stock: Bring a large stockpot of salted water to a rolling boil; cook lobsters, 5 minutes. Remove lobsters, reserving water. When cool enough to handle, remove the tails and claws, keeping shells intact, and set aside for the paella. Place lobster bodies on a sturdy surface, belly side up. Cut in half, then quarters, keeping shells intact.In a medium stockpot over medium-high heat, add olive oil and sauté lobster body pieces, 5–7 minutes. Add tomato paste, stirring to thoroughly coat the lobster. Deglaze the pan with brandy and cook off the alcohol, about 1 minute; then add onion, carrots, tomato, fennel, and garlic and continue sautéing, 3–5 minutes. Add 6 cups of the reserved lobster water, stir to combine, cover, reduce heat to medium-low, and simmer for 2–3 hours. Strain stock into a large bowl, discarding lobster pieces and vegetables. Remove 1 cup of the stock; steep a pinch of saffron threads in the hot liquid for 3–4 minutes, then return the saffron-infused broth to the stock. Reserve Lobster Stock for paella. Sofrito: Dice onions and bell pepper (pieces should be the same size); set aside in separate bowls. Cut tomatoes in half lengthwise. Grate over a bowl flesh-side down and discard tomato skin. Smash garlic cloves and roughly chop; use the back of the knife to crush the garlic, then continue to chop to a rough paste. Set aside. Discard stem and seeds of soaked chile; use a knife gently scrape the flesh from the skin of the pepper. Discard the skin and set flesh aside in a small bowl.
In a large, heavy-bottomed sauté pan, heat 1 tablespoon olive oil over medium-high heat. Add the onions, then 1 more tablespoon olive oil. Stir and sweat until translucent, about 5 minutes.
Add the bell peppers and chile to the onions; then add the grated tomato and season with salt and pepper. Stir, then add garlic and vinegar. Stir to thoroughly combine all ingredients. Reduce heat to low and simmer Sofrito uncovered, stirring occasionally until reduced by two-thirds, about 2 hours. Makes 2 cups Sofrito. (Note: Freeze leftovers in small portions to defrost and use as needed. Even for a quick rice or pasta dish, a bit of this will make all the difference in flavor.)
Paella: Discard outer casing from chorizo, then dice and set aside. Prepare lobster pieces by cutting across the tail in 2-inch medallions, leaving shells intact. Use the back of a knife to crack the claws. Set lobster aside. Place shrimp and chicken on a rimmed baking sheet and season with salt and pepper. Set aside.
In a paella pan over medium-high heat, add 2 tablespoons olive oil and heat until smoking, 1 minute. Lay chicken pieces skin-side down, then the shrimp, in an even layer to sear. After 30 seconds, turn shrimp, cook 30 more seconds, then remove and set aside. (Shrimp will be lightly seared, not fully cooked.)Continue cooking chicken until the skin is golden brown, 2 minutes; to maintain even heat, use the handles to turn the pan as the chicken cooks. Turn chicken over and add chorizo. Add 1 cup Lobster Stock; it should sizzle and steam when it hits the hot pan. Add 1 cup Sofrito and gently stir to combine and return to active simmer, about 30 seconds.
When paella is at an active simmer, add rice, distributing evenly in the pan. (Note: An active simmer is important; it helps the rice release its starch, giving paella the proper consistency.) Add another 1½ cups stock; gently stir, and season with a pinch or two of salt. Let simmer, turning the pan as necessary so the heat is evenly distributed. Lower heat to medium-low and continue cooking the paella, adding ½ cup of stock at a time as it absorbs, about 20 minutes.
Flip the chicken pieces, adding stock to the parts of the pan that look drier. As the paella cooks and the dish firms up, the rice will begin to change from translucent to opaque. Stir gently to allow a bit of socarrat (the crispy bottom that is the prized part of paella) to begin to form on the bottom.
When the rice has nearly finished cooking (about 25 minutes total), bury the clams into the paella, followed by the mussels. Then layer the shrimp on top. Raise the heat to medium-high.
Add lobster tail and claw pieces to the pan. Flip the shrimp to let them cook completely through. Check that the chicken is cooked through: Cut into a piece and check that the color is rosy, but not deep pink. Taste the paella for seasoning and add a pinch more salt, if necessary. Finish the paella with a final drizzle of olive oil. Remove the paella pan from the heat and let rest for 5–10 minutes before serving. (Note: Discard any clams or mussels that have not opened.)