Prepare the shellfish

In an 8- to 10–quart covered stockpot bring 1 inch of well-salted water to a boil.

Add the lobster to the pot and steam, covered, until it is bright red and a small leg easily pulls away from the body, 6 minutes per pound, plus 2 minutes for each additional 1/4 pound. Using tongs, transfer the lobster to a platter and let sit at room temperature until cool enough to handle.

Twist off the tail, claws, and legs. Remove and rinse the tomalley from the body cavity and discard the roe if present; reserve the body. Extract the meat from the claws and tail. Reserve the legs and all the shells. Slice the tail meat in half lengthwise and remove and discard the dark vein that runs along the inside. Refrigerate meat and shells until ready to use.

Debeard and scrub the mussels. Soak in a couple of consecutive salt-water baths to remove any grit. Lift the mussels from the water and rinse a final time. Refrigerate until ready to use.

Remove and reserve the shells from the shrimp, leaving the tails intact, and devein the shrimp. Refrigerate the shrimp and shells until ready to use.

Make the sofrito

Cut the tomatoes in half crosswise and use your fingers to remove and discard the seeds. Grate the cut sides of the tomatoes on the large holes of a box grater over a plate. Discard the remaining skins. On the same plate, do the same with the red pepper. Finely grate the garlic with a rasp grater; add two-thirds of the garlic to the red pepper and tomato and reserve the rest.

On a separate plate, grate half of the onion on the large holes of a box grater.

Heat 3 Tbs. of the oil in a 12-inch paella pan or oven-safe skillet over medium-low heat until shimmering. Add the grated onion and stir occasionally until it begins to soften but not brown, about 2 minutes. Add the tomato-pepper mixture, pimentón, and 1/2 tsp. salt and stir to combine.

Bring to a simmer and cook until the mixture turns a darker shade of red, about 30 minutes. If the mixture begins to stick, lower the heat and loosen with a few tablespoons of water, scraping any brown bits into the mixture with a wooden spoon. Continue to stir frequently, pushing the mixture into the center of the pan. Season to taste with salt and pepper.

Make the shellfish stock

Coarsely chop the remaining half of the onion. Heat 1 Tbs. of the oil in a 5- to 6-quart saucepan over high heat until shimmering. Add the shrimp shells and cook, stirring, until they turn pink, about 1 minute. Add the onion, carrot, leek, and fennel and cook, stirring occasionally, until they begin to soften, about 5 minutes.

Add 4 cups of water and 1/2 cup of the clam juice to the pot. Break up the reserved lobster body and add it to the pot with the other lobster shells and legs. Bring to a boil and reduce to a simmer for 10 minutes, stirring occasionally while continuing to break up the shells with a wooden spoon.

Meanwhile, in a 1-quart saucepan over low heat, lightly toast the saffron until it becomes fragrant and slightly darker, about 1 minute. Pour in the remaining 1/2 cup clam juice and bring to a boil. Remove from the heat, cover, and set aside to infuse, about 10 minutes.

Remove the fish stock from the heat and strain into a fine-mesh strainer, pressing down on the solids with a ladle or wooden spoon to extract as much stock as possible. Discard the solids. Add the infused saffron mixture to the fish stock and stir well to incorporate. Season to taste with salt and pepper.

Make the paella

Position a rack in the center of the oven and heat the oven to 400°F.

Heat the pan of sofrito to medium high, add the rice, and stir with a wooden spoon for about 2 minutes.

Spread the rice evenly with the spoon. Slowly add 3 cups of the fish stock to just cover the rice. Shake the pan gently to even out the layer of rice. From this point on, do not stir the rice. Bring the stock to a boil and cook until the rice plumps up and absorbs most of the liquid, about 6 minutes.

Nestle the mussels evenly over the rice. If there are dry-looking patches of rice, add a bit of the stock. Transfer to the oven and bake for 5 minutes. Rotate the pan and sprinkle the sugar snap peas over the rice. Bake for another 5 minutes. Taste the rice at this point: It should be slightly al dente and the mussels should be open. If the rice is still hard, add a little more broth and bake for another 2 minutes and taste again. Remove from the oven, cover with a clean kitchen towel and let rest for 15 minutes.

Meanwhile, heat the remaining 1-1/2 Tbs. of the oil in a 10-inch skillet on high heat until shimmering. Add the shrimp and cook, stirring, until lightly seared, about 1 minute. Add the lobster tails and claw meat, the reserved garlic, 1/4 tsp. salt, and 1/4 tsp. pepper, and continue to cook until the shrimp turn pink, about 4 minutes. Remove from the heat and let rest until the paella is ready, about 5 minutes.

Remove the towel from the paella pan and arrange the shrimp and lobster meat evenly over the rice. Pour any remaining liquid from the skillet over the shrimp and lobster meat. Serve immediately with lemon wedges.

Make Ahead Tips

You can make the fish stock and the sofrito ahead (do not add the rice to the sofrito).