How to Make It
In a large pot, combine the octopus with all of the remaining ingredients and add enough water to cover by 1 to 2 inches. Bring to a boil, reduce the heat to moderate and simmer until the octopus is tender, about 45 minutes. Let the octopus cool in the liquid, then drain and pat dry. Separate the head and tentacles and discard the hard beak. Thinly slice the octopus.
In a large skillet, heat 2 tablespoons of the olive oil. Add the finely chopped onion, cover and cook over moderately low heat, stirring frequently, until softened, about 5 minutes. Add the chopped red pepper along with the cubanelle peppers, yellow pepper, carrot, tomato, capers, garlic and bay leaf. Cover and cook over moderate heat, stirring frequently, until the peppers are softened, about 5 minutes. Add the tomato juice, dissolved tomato paste and the cilantro and cook partially covered over low heat, stirring occasionally, until thickened, about 30 minutes. Set aside 1/2 cup of the Creole sauce and discard the bay leaf.
Add the sliced octopus to the Creole sauce in the skillet and simmer for 10 minutes. Season with salt and let cool completely. Scrape the octopus filling into a bowl and refrigerate until chilled, at least 30 minutes.
Meanwhile, preheat the oven to 425°. Cut the remaining onion half into wedges and place in a small roasting pan. Add the quartered red pepper and the hot red chiles and toss with the remaining 1 tablespoon of olive oil. Roast for about 20 minutes, until the vegetables are softened and blistered in spots. Transfer the vegetables to a blender and let cool. Add the reserved 1/2 cup of Creole sauce and puree until smooth. Add the mayonnaise and lime juice and pulse to blend. Season the sauce with salt and refrigerate until chilled.
5. make the dough In a standing electric mixer fitted with the paddle, mix the 3 1/2 cups of flour with the salt and vegetable shortening at low speed until evenly combined. Add the egg and ice water and mix until the dough is evenly moistened. Knead the dough briefly until smooth, then divide it into 16 equal pieces. Roll each piece into a ball and transfer to a lightly floured baking sheet. Cover with a lightly moistened paper towel and plastic wrap and let stand at room temperature for 30 minutes.
On a barely floured surface, roll out each ball of dough to a 6-inch round. Spoon about 1/4 cup of the octopus filling onto each round. Fold the dough over the filling, press out any air bubbles and pinch to seal. Using a round pastry wheel to trim the pastries, cut off any excess dough. Transfer the turnovers to a lightly floured baking sheet.
In a large, deep skillet, heat the vegetable oil to 350°. Cover a rack with paper towels and set it on a baking sheet. Working in batches, fry the turnovers over moderately high heat, turning once, until golden, about 4 minutes. Drain on the paper towel–lined rack. Let cool slightly, then serve the turnovers with the Creole mayonnaise.