This is my attempt to capture the essence of one of America's greatest sandwiches: the muffuletta from Central Grocery on Decatur Street in New Orleans. I mimic the oily richness of the bread with homemade focaccia, the brine of the olive salad with olives freshened with citrus zest, and the brightness of the giardiniera with pickled carrots. The richly spiced tasso ham speaks not only to the smokiness of the cured meats in the original but also reminds me of the Creole culture common in that part of the country. (If you cannot find tasso, other pork cured meats such as mortadella or salami can be substituted.) Finally, the melted mozzarella and provolone may be an improvement on an already perfect sandwich.
To make the focaccia, in a bowl, stir together the flour, salt, sugar, yeast, and rosemary. Add the water and canola oil and, using a wooden spoon, stir until the dough comes together in a rough, shaggy ball. Lightly flour a work surface and turn the dough out onto it. Knead for about 15 minutes, until smooth and elastic, then shape it into a ball.
Oil a large bowl with canola oil, transfer the dough to the bowl, turn the dough to coat evenly with the oil, cover the bowl with a damp kitchen towel, and let the dough rise in a warm spot for 20 minutes. Meanwhile, preheat the oven to 450 degrees F. Oil a 9 1/2- by 13-inch sheet pan with 1 tablespoon of the olive oil.
Punch down the dough and transfer it to the prepared pan. Using your fingers, press the dough evenly over the bottom of the pan. Let rise in a warm spot for an additional 20 minutes. Brush the top with the remaining 1 tablespoon olive oil.
Bake for about 15 minutes, until golden brown. Let cool completely in the pan on a wire rack, then cut into 4 even squares.
To make the carrots, put the carrots in a heatproof bowl. In a small saucepan, combine the vinegar, salt, and sugar and bring to a boil over high heat, stirring to dissolve the salt and sugar. Pour over the carrots and let cool.
To make the olives, in a small bowl, combine all of the ingredients and mix well.
Split each focaccia square in half horizontally. Line up the 4 bottom halves, cut sides down, on a work surface. Top each half with 3 provolone slices, followed by one-fourth each of the olives, carrots, basil, tasso, and mozzarella. Close the sandwiches with the top halves of the focaccia squares, cut sides up.
Line a large platter with paper towels. In a skillet over high heat, melt 1 tablespoon of the butter. Turn down the heat to low and add 1 sandwich. Press down on the top of the sandwich with a spatula to flatten it so it cooks evenly, then cook, turning once, for 2 to 3 minutes on each side, until browned and crisp on both sides and the cheese is melted. Transfer to the prepared platter to blot the excess grease. Repeat with the remaining butter and sandwiches.