The round, flat shape of the muffuletta, the bread this sandwich is made on, was perfected by the Italian immigrants in southern Louisiana. Traditionally, a mixture of Italian meats and cheeses are piled high on the bread, and the olive oil brings it all together. When served warm, the gently melted cheese makes this sandwich simply irresistible.
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2. For the bread: Whisk the buttermilk, milk, sugar and yeast in a medium bowl and set aside until bubbly, about 2 minutes. Whisk in the salt, baking soda and egg. Add 2 1/2 cups of the flour and whisk until well combined. Switch to a rubber spatula and add the remaining 1 1/2 cups flour, 1/2 cup at a time, stirring until well combined after each addition. Lightly flour a work surface. Turn the dough out onto the floured surface and knead 10 to 12 times just to bring it together, and then form it into a ball. Lightly oil a medium bowl. Turn the dough in the bowl until coated, cover with a towel and set aside to rise until doubled in size, about 40 minutes.
3. Preheat the oven to 375 degrees F. Line 2 rimmed baking sheets with parchment.
4. On a lightly floured surface, divide the dough into 6 pieces (about 5 ounces each), form each into a ball and place 3 on each of the prepared baking sheets. Brush with milk and sprinkle with sesame seeds. Cover with a towel and allow to rise until nicely puffed, 25 minutes.
5. Bake until golden brown, 15 to 20 minutes. Allow to cool completely on the baking sheets.
6. For the sandwiches: Cut the breads in half horizontally and brush the bottoms with some of the oil from the olive salad. Divide the ham, mortadella, salami, provolone and Swiss cheese among the sandwiches. Top with a generous amount of olive salad, at least 1/3 cup, and top with the other bread half. Wrap tightly in plastic wrap and place on a rimmed baking sheet. Place another baking sheet on top and then put something heavy, such as a cast-iron skillet, on top of that to gently press the sandwiches together. Press at least 15 minutes and up to an hour. To serve, slice the sandwiches in halves or quarters.