Cook the lobsters (if you purchased steamed whole lobsters, skip to step 4.): Add wine, onion, carrot, celery, parsley, lemon, bay leaf, and peppercorns to a large Stainless Clad Stock Pot. Add enough water to submerge lobsters in the cooking liquid, about 16 cups. Cover and heat over high until simmering. Remove lid, reduce to medium-low, season generously with salt, and continue to cook until flavorful, about 15 minutes.

To kill the lobsters just before steaming, place the tip of a Chef Knife behind the lobster’s eyes, just below where the claws meet the body, and quickly plunge the knife all the way through the head. The legs may continue to move a bit afterwards.

Bring cooking liquid to a rolling boil and add the live lobsters. Cover and boil until lobsters are bright red and tails are curled, about 11 minutes. Transfer lobsters to a Sheet Pan to cool. Strain cooking liquid into a medium Stainless Clad Saucepan and reduce over medium-high heat until reduced roughly by half while you complete the additional steps (this is the lobster stock you will need for the sauce).

Once lobsters are cool enough to handle, remove claws and crack and remove meat. Use kitchen shears to split lobsters lengthwise from head to tail. Remove coral and tomalley and discard or reserve for another use. Remove tail meat and chop claw and tail meat into 1” pieces. Set aside.

Make the sauce: Add 3 Tbsp butter to a small or medium Stainless Clad Saucier. Heat over medium until butter is melted and foaming. Add flour and whisk constantly, until roux is foaming and light blonde in color, adjusting heat if necessary, 2–3 minutes. Slowly whisk in 1 cup reduced lobster stock (from step 3). Bring to a low boil, whisking until thickened (this is velouté sauce!). Remove from heat and allow to cool slightly.

In a medium bowl, whisk together cream, egg yolk, mustard, and paprika. Slowly pour velouté sauce into the bowl, whisking constantly to temper and avoid over cooking the egg. Return sauce to Saucier and stir with a wooden spoon while gradually increasing the heat to medium. Bring to a low boil and cook about 2 minutes, until the sauce is thick enough to coat the back of the spoon. Reduce heat to low, gradually add Parmesan to sauce, and whisk until smooth with each addition. Taste and adjust salt, mustard, and paprika to your taste. Keep warm.

To finish the dish: Heat a medium Stainless Clad Frying Pan over medium high. Add mushrooms and cook, stirring and flipping often, until softened and liquid has evaporated, 3—5 minutes. Add 1 Tbsp. butter and shallot, and season with salt. Cook, stirring, until mushrooms start to brown and shallot is softened, about 2 minutes. Add remaining 1 Tbsp. butter, garlic, and chopped lobster. Stir to combine and cook until garlic is fragrant, about 1 minute. Add cognac and cook until reduced and alcohol taste is cooked off, 2–5 minutes (you can flambé here if you feel comfortable). Do not overcook lobster. Stir in lemon juice to your liking. Taste and adjust seasoning.

Preheat broiler with rack 6 inches from heating element. Fill lobster cavities with lobster mixture. Spoon sauce over top. Top with additional Parmesan and paprika and broil until top is light golden brown, 2–4 minutes. Top with chopped parsley and serve with lemon wedges and extra sauce alongside.