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For the stuffing: Heat the butter in a pan over medium heat. Add the fennel and onions and cook until soft. Add the bread crumbs, dill and lemon juice and toss to combine. Add the shrimp and cook until pink, 2 to 3 minutes.
Heat the oil in a pan over medium-high heat. Sear the fish on each side until golden brown and the skin is crisp, 3 to 4 minutes. Transfer the fish to a parchment-lined baking sheet and roast until a pairing-knife stuck into the center of the fish comes out warm to the touch, about 10 minutes.
Serve the fish with the lobster sauce.