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Bake the potato until tender, about 45 minutes to 1 hour.
Spread about a 2-foot sheet of foil on a flat surface with 1/2 of it hanging off the counter in front of you. Put the fish, skin side down, horizontally, onto the foil near the end of the counter leaving about 1 1/2-inches of foil on both sides. Season the flesh side of the fish with salt and pepper, to taste, and sprinkle with some dill, parsley and a few lemon slices. Drizzle about 2 tablespoons oil on top of the fish. Fold the foil back over the fish and fold in the sides of the foil twice. Leaving some air in the package, roll the top of the foil down twice gathering the top to make a window , leaving about 6 to 8 inches between the fish and the top. It should look like a small package, with all the sides sealed. There should be enough room left around the fish to allow for the steam to build up as it cooks, creating an inflated envelope around the fish. Repeat with remaining 3 pieces of fish.
Put the garlic cloves in the bowl of a food processor and pulse until the garlic is finely chopped. Add the egg yolks and pulse to blend. When the potato is completely cooked and still warm, put it on a flat surface and split it open lengthwise. Use a tablespoon to scoop out the flesh, totaling about 3 to 4 tablespoons, and add it to the processor. Pulse to combine. With the processor running, slowly add the 1/2 cup of olive oil, through the opening at the top, in a steady stream. Add the sugar and the lime juice and pulse to incorporate. Season, to taste, with salt and pepper. Cook s Note: If the mixture is too thick, add a little cold water to loosen the texture. Transfer the mixture to a serving bowl, cover and refrigerate.
Using 2 baking sheets, put 2 of the foil envelopes on each sheet and add a little water to the bottom of the sheet pan. This water will create additional steam in the oven as the fish cooks. Put the baking sheets on 2 shelves of the oven and bake, undisturbed, until the envelopes begin to puff, about 12 to 15 minutes. Remove from the oven and immediately transfer to 4 dinner plates. To serve, break open the foil and drizzle with the aioli.