In a double boiler over medium heat, gently melt the shortening with the milk and honey; let cool slightly. Add yeast and let sit until foamy, 8 to 10 minutes. Lightly beat the egg in a small bowl and set aside. In yet another bowl, gently sift together the flour, baking soda, and salt.
On a large wooden cutting board or counter, dump the flour mixture in a mound and create a well in the center. Pour the egg and half of the shortening mixture into the well and combine the mixture with your hands, incorporating the flour slowly. Keep kneading once all the flour is incorporated; the dough should hold together after a few strokes. Gradually add the rest of the shortening mixture one tablespoon at a time.
Knead dough for 15 minutes; then let it rest for another 15 minutes between two plates. While the dough is resting, pre-heat a griddle or flat-top and grease it, sparingly, with shortening.When the dough is rested, break off about one cup of dough, enough to form a ball about the size of your fist. On an unflowered surface, roll out the dough to form thin, 10'' rounds. Repeat with remaining dough.When the griddle is hot, cook the rounds, about three minutes on each side, or until dark brown spots appear.
When the piadina are cooked through, remove them from the griddle and slice in half. Fill the sandwich with prosciutto, cheese, and arugula and serve while still warm.