- Make the pesto: In a medium bowl, pour boiling water over chiles. Cover bowl with a sheet of plastic wrap and set aside for 15 minutes. Transfer chiles to a large paper towel-lined plate. Reserve ½ cup of chile water and discard the rest.
- In a small skillet over medium heat, warm 2 tablespoons olive oil and add hazelnuts. Stir frequently, until the hazelnuts are golden brown and fragrant, about 5 minutes. Transfer hazelnuts to a blender or food processor and add the softened guajillos, ¼ cup of chile water, 1/3 cup olive oil, and garlic. Process until pesto is smooth, using the reserved ¼ cup of chile water if needed to thin and smooth mixture. Scrape mixture into a bowl and fold in grated cheese. Adjust seasoning with salt and pepper to taste.
- Make the lobster: Bring a large, heavy pot of water to a boil. Add the lobsters and cook for 5 minutes. Transfer lobsters to two large baking sheets. When the lobsters are cool enough to handle, twist off tails and claws. Using scissors, cut down the center of the tail shells and remove tails in one piece. Cut down the length of each tail and discard the dark intestinal veins. Crack claws and knuckles to remove the meat in one piece. Cover and refrigerate lobster meat.
- Make the sauce: In a large pan over medium heat, sauté garlic, carrots, fennel, and onions in ¼ cup of the olive oil, stirring occasionally, until softened, about 8 minutes. Add cognac and cook until almost evaporated, about 3 minutes. Cover with tomatoes and clam juice, and bring to a boil. Simmer over medium heat for 15 minutes.
- Make the pasta: In a large, wide pan, heat the remaining ¾ cup olive oil until shimmering. Add pasta and cook over moderately high heat, stirring constantly, until browned, about 3 minutes. Stir in half the tomato-clam juice mixture, cover, and cook for 3 minutes. Add one ladle of remaining sauce and stir until absorbed into pasta. Repeat until pasta is al dente. Season with salt, pepper, and a squeeze of lemon to taste.
- To assemble: Preheat oven to 350 degrees. On a baking sheet, drizzle cooked lobster with melted butter and place in oven for 4 minutes to warm through. Spoon the fideos on a platter, top with warmed lobster, and serve with guajillo pesto.