MAKE AHEAD: The slaw can be refrigerated for up to 5 days.
For the slaw
• 1 pound (about half of a medium-size head) green cabbage, shredded or finely chopped (about 6 cups)
• 1/2 cup sugar
• 1 to 2 teaspoons kosher salt
• 1/4 teaspoon celery seed
• 1/4 cup vegetable oil
• 1/4 cup apple cider vinegar
• Freshly ground black pepper
For the twice-fried potatoes
• 6 to 8 large (4 to 5 pounds) russet potatoes, washed well
• 8 cups vegetable oil, for frying
• Kosher salt
For the meat and cheese
• 3 tablespoons unsalted butter
• 2 pounds spicy, thinly sliced capicola ham
• 8 thin slices provolone cheese (about 5 ounces)
• 4 vine-ripened tomatoes, cut into 16 thin slices
• 16 large slices of soft Italian bread (18 ounces total)
For the slaw: Combine the cabbage, sugar, salt and celery seed in a colander set over a medium bowl. Let stand at least 1 hour and up to 4 hours; the cabbage will be wilted (about 4 cups total).
Discard the draining liquid in the bowl; rinse and dry the bowl, then transfer the wilted cabbage to the bowl.
Add the oil and vinegar; toss to coat. Season with pepper to taste. Cover and refrigerate until ready to serve.
For the twice-fried potatoes: Preheat the oven to 200 degrees. Line a few large baking sheets with several layers of paper towels. Fill a large bowl with cold water. Cut the (unpeeled) potatoes lengthwise into 1/4-inch-thick sticks. Submerge in the cold water. Rinse in subsequent changes of cold water to remove all visible starch, then drain in a colander and spread the
potatoes on the paper towels, patting the potatoes dry.
Heat the oil in a large, heavy pot over medium-high heat, until the oil temperature reaches 320 degrees.
Fry the potatoes in 4 batches; each batch will take 2 to 4 minutes. Stir occasionally as they cook, until the fries are soft and cooked through but still pale. Allow enough time for the oil to return to 320 degrees between
batches; use an instant-read thermometer to monitor the oil. Use a slotted spatula to transfer the potatoes to the lined baking sheets.
Increase the heat to high (or as needed) so that the temperature of the oil reaches 375 degrees. Preheat the oven to 200 degrees. Refresh the paper towels on the baking sheets as needed.
Cook the fries a second time, working in 4 batches; each batch will take 2 to 3 miinutes, until the fries are crisp and golden brown. Transfer to the lined baking sheets. Immediately season lightly with salt, then place in
the oven to keep the fries warm.
For the meat and cheese: Melt the butter in a heavy skillet over medium heat. Have ready a large baking sheet.
Separate the ham slices and add to the skillet, turning them as needed until the slices are warmed through.
Transfer the slices to the baking sheet, creating 8 equal portions. Top each with a slice of provolone cheese.
Place in the oven (along with the fries) just until the cheese has melted.
For assembly: Place the portions of cheese-topped ham on 8 bread slices. Top with a large handful of the warm fries, then pile about 1/2 cup of the slaw on each portion. Garnish with 2 tomato slices for each portion;
use the remaining 8 pieces of bread to finish each sandwich. Serve warm.