For the spread: In a small bowl, whisk together the mayonnaise, Calabrian chili paste, lemon zest and salt. Set aside.
For the sandwich: Heat a large skillet over medium-high heat. Add the oil and sausage and cook, breaking the sausage into large pieces with a wooden spoon, until golden brown, cooked through and crispy in parts, about 15 minutes. Drain off excess fat if necessary.
Meanwhile, in a medium bowl, combine the arugula, basil, cider vinegar and salt. Toss well to coat.
Split the rolls in half horizontally, leaving one side intact. Open the rolls like a book and remove and discard some of the inside of the bread. Spread 1/3 cup of the spicy mayonnaise evenly on each roll. Fill the bottom half of each roll with the crispy sausage mixture. Fill the top half of each roll with 3/4 cup of the giardiniera. Divide the dressed arugula between the sandwiches and close them, pressing gently on the top. Cut each sandwich in half and serve.