Preheat the oven to 350°. Spread a 1/2-inch layer of salt in a small baking dish. Arrange the shallots on the salt and bake for 45 minutes, until soft. Let cool. Peel the shallots and transfer them to a blender. Add the cream, lemon juice, mustard and thyme and puree. Transfer the shallot cream to a saucepan and season with salt and pepper.
In a large nonstick skillet, heat the oil. Add the capers and cook over moderately high heat until crisp, 2 minutes. Using a slotted spoon, transfer the capers to a plate. Season the tuna with salt and pepper and spread with the mustard. Heat the oil left in the skillet until shimmering. Add the tuna and cook over high heat, turning once, until browned outside and medium-rare within, 2 minutes per side.
Transfer the tuna to plates. Gently heat the shallot cream and spoon over the fish. Scatter the capers on top. Serve with the Eggplant Relish and steamed green beans.
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