How to Make It
In a standing mixer fitted with the paddle, mix the flour with the baking soda and salt. Add the shortening and beat at low speed until the mixture resembles coarse meal. Gradually beat in the water on medium-low speed until the dough is smooth, about 3 minutes. Divide the dough into 8 pieces, cover with plastic wrap and let stand at room temperature for 30 minutes.
Preheat the oven to 400°. In a medium roasting pan, toss the butternut squash with the sliced pancetta, the 3 tablespoons of olive oil and a pinch each of salt and pepper. Roast for about 25 minutes, stirring occasionally, until the squash is tender and browned in spots and the pancetta is browned.
On an unfloured surface, roll out each piece of dough to an 8-inch round, about 1/8 inch thick. Heat a griddle or cast-iron skillet. Cook the piadina one or two at a time over moderate heat, turning once, until lightly charred in spots and no longer doughy, 3 to 4 minutes. Transfer to a shallow bowl and cover with foil to keep warm while you cook the rest.
Arrange the piadina on a work surface. Top 4 of the piadina with the butternut squash–pancetta mixture and shaved pecorino and fold to close. Top the other 4 with the prosciutto, stracchino cheese and arugula. Drizzle lightly with olive oil, season with salt and pepper and fold to close. Serve the piadina immediately, either whole or cut into wedges.