In a 2-gallon saucepot, bring 1½ gallons lightly salted water to a boil. Plunge lobsters, claws first, into the boiling water and cook until tail floats, 10–12 minutes. Remove from water and cool. Reserve water for boiling pasta. Once lobsters are cool, remove tail, claw and knuckle meat from shell and set aside, discarding shells. Devein lobster tails then slice into ⅛-inch medallions. Dice claw and knuckle meat then set aside. In a 2-quart saucepot, heat bacon and oil over medium-high heat. Cook 10–12 minutes or until fat renders and bacon browns, stirring occasionally. Drain and reserve 1 tablespoon bacon fat then set aside. Add garlic and cook until softened and edges turn golden brown, taking care not to burn. Add heavy whipping cream and bring to a rolling boil then reduce to simmer, stirring often. Cook until cream mixture is reduced to ¼ cup then remove from heat and keep warm. Meanwhile, cook pasta according to package directions. Drain and place pasta into a large bowl. Set aside. In a medium bowl, whisk together reserved bacon fat, Pecorino Romano cheese, eggs, egg yolk, salt and pepper until ingredients are well combined. Slowly whisk hot cream mixture into Pecorino mixture. Gradually pour Pecorino and cream mixture over spaghetti, tossing to coat. Add lobster meat and toss to combine. Let pasta rest 2–4 minutes or until sauce has thickened slightly and coats spaghetti. Serve immediately.