Peel the onion, shallots and carrots and cut into small chunks. Cut the tomatoes and celery into similar sizes. Peel the garlic cloves and roughly chop.
Break down the lobsters and cut the heads in half; keep the roe in to add flavor. Add 2 tablespoons oil to a large, high-sided skillet over medium-high heat and allow the oil to heat. Add the lobsters, onions, carrots and half of the celery to the skillet and saute until the onions start to become translucent, about 10 minutes. Add stock to cover and bring to a boil, then lower to a simmer. Add the brandy, cook for 5 minutes, then strain the stock and set aside the lobster pieces with flesh inside. Remove meat from lobster shells and chop into 1-inch chunks.
Heat 2 tablespoons oil in a large skillet over high heat. Toss the potatoes with salt and pepper and add to the skillet. Flip occasionally to avoid burning and remove once they are par-cooked, about 8 minutes. Set aside.
Heat remaining tablespoon oil in a large skillet over medium-high heat, then add the shallots and remaining celery and saute until translucent, about 5 minutes. Add the garlic and cook until fragrant, 2 more minutes. Add the tomatoes, potatoes and lobster, season with salt and pepper and cook until the tomatoes start to break down. Flambe with vermouth and cognac. Once the fire dies down, stir in the tarragon and heavy cream.
This recipe has been updated and may differ from what was originally published or broadcast.