Bring a large pot of water to a boil. Place the lobster claws in water for 3 minutes. Dont cook them through, but enough to be easily removed from the shell. Remove the claws from the water. Add the tails to the water and cook for 3 minutes. Extract the meat from the claws and the tails and set aside.
In a large saute pan over medium-high heat, add 2 tablespoons olive oil. Add the shallots, garlic and any scraps from the sliced fennel, saute until slightly brown. Add the lobster bodies, the cracked shells from the tails and the claws, saute another 5 minutes. Add a few good pinches of kosher salt and stir.
Deglaze with vermouth and simmer until the alcohol is mostly cooked down. Add the heavy cream, turn the heat to low. Let simmer for about 30 minutes. Strain, add tarragon and pepper. Dice the par-cooked lobster tail, and add to the cream mixture.
While the cream is simmering, crack the eggs into a large bowl and whisk. Add the sliced fennel and coat well. Pour the bread crumbs onto a plate. Dip each fennel slice in the panko, pressing firmly to coat completely on both sides. Lay the coated fennel slices on a plate.
In a large saute pan, heat 1/2 cup of canola oil over medium heat. Fry the breaded fennel one piece at a time, making sure not to overcrowd the pan. When golden brown on one side, flip and cook the other side to the same golden brown, about 3 to 4 minutes on each side. Drain the fennel on paper towel and season with salt.
The next step works better with a grill, but if this isnt an option, a saute pan works well.
Preheat a grill. Coat the par-cooked claws with olive oil, salt, and pepper. Place the claw meat on a very hot grill, cook about 2 minutes per side, and remove from the heat.
To construct a plate, lay down one slice of fennel, then place one claw on top. Layer another slice of fennel and another claw on top. Spoon the cream liberally over the top, making sure each plate gets a nice amount of tail meat.
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