How to Make It
Using a mandoline, cut the carrots into thin strips. In a small skillet, heat 1/4 cup of the olive oil. Add the carrots, shallots, thyme sprigs, bay leaves and garlic and cook over moderate heat until the vegetables are softened, about 6 minutes. Remove from the heat. Stir in the vinegar and season with salt and pepper. Let stand for 10 minutes.
In a large nonstick skillet, heat 1/4 inch of olive oil until shimmering. Season the mackerel fillets with salt and pepper and add them to the skillet skin-side down. Cook over high heat until the skin is browned and crisp, about 4 minutes. Turn and cook the fillets for 1 minute longer, until almost cooked through. Transfer the fillets to a large, shallow glass or ceramic dish, skin side up. Top with the vegetables and their marinade. Let stand at room temperature for 3 hours, occasionally spooning the marinade over.
Discard the thyme sprigs and bay leaves. Arrange the mackerel on plates and spoon the vegetables and their marinade on top. Dollop with the tapenade and serve.