Preheat oven to 325°. Break spaghetti into 1-inch pieces; toss with 1 1/2 teaspoons olive oil. Spread spaghetti in a single layer on a rimmed baking sheet. Bake 8 to 10 minutes or until toasted and golden brown.
While spaghetti bakes, combine wine, bay leaf, and 6 cups water in a large stockpot over high heat; bring to a boil. Add lobsters; cook 8 minutes. Remove lobsters from pan with tongs, reserving cooking liquid. Remove lobster meat; return lobster shells to pot over medium heat. Simmer 10 to 15 minutes; strain, reserving liquid.
Preheat broiler to high. Heat remaining 1 1/2 teaspoons oil in a large paella pan or wide enameled pan over medium-high heat. Add garlic and paprika; cook 1 minute, stirring constantly. Add toasted spaghetti and 2 cups reserved cooking liquid to pan; cook until liquid is absorbed, stirring constantly. Add remaining reserved cooking liquid to pan. Bring to a boil; cook until liquid is reduced by half.
Transfer pan to oven; broil 6 inches from heat 8 to 10 minutes or until liquid is absorbed and edges of spaghetti are crispy. Remove pan from oven, and arrange lobster pieces on top to warm through. Sprinkle evenly with chives; serve with Classic Aïoli and lemon wedges.