To make the piadina dough, in the bowl of a stand mixer fitted with the dough hook, combine the flour, baking soda and salt. Add the shortening and mix until blended through and the mixture looks like fine sand. Stir in the warm water until the dough comes together. Knead on high for about 5 minutes until it pulls away from the sides of the bowl and is smooth and not sticky. If the dough doesn't come together, add a small amount of water more a tablespoon at a time, until it does. Cover with a kitchen towel and allow to rest at room temperature for 30 minutes.
Preheat oven to 225 degrees F. Heat a cast-iron skillet over moderate heat until very hot.
Cut the dough into 8 equal pieces and keep covered with the kitchen towel. Working with one piece at a time, roll out on an unfloured work surface into an 8-inch round, about 1/8inch thick. Brush both sides with olive oil and cook in preheated pan, turning once, until golden and cooked through, about 3 to 4 minutes. Transfer to a baking sheet and cover with a clean kitchen towel. Keep warm in oven while you cook the rest.
In a small bowl, combine the ricotta, 1 tablespoon lemon juice, 1 tablespoon olive oil and a pinch of salt and pepper. Set aside.
In a separate bowl, combine the arugula, remaining tablespoon lemon juice and olive oil and season with a pinch of salt and pepper. Set aside.
To assemble the piadine, fold each in half and then fill with ricotta and/or mozzarella, arugula and prosciutto. Top with a bit of coarse salt, if you wish. Serve immediately. Enjoy!