Thaw the puff pastry on your counter at room temperature for 30-35 minutes. The pastry should unfold easily but still be cold.
Prepare the filling while the pastry thaws: Heat 1 tbsp olive oil in a skillet over medium heat. Add the shallots to the skillet along with a pinch of salt and some fresh cracked pepper. Sauté for 3 to 4 minutes until they begin to soften. Add the Italian herb blend and the arugula, in batches if necessary, and stir as it wilts. As soon as the arugula has just wilted, remove it from the heat.
Remove the arugula to a paper towel to drain and cool. Once it has cooled enough to handle, press the arugula with another paper towel to remove excess moisture. Set aside until you're ready to fill the dough.
Preheat the oven to 400°F.
Lightly dust a large work surface with flour (or work in batches if your work space is small). Set out the 2 squares of pastry leaving a few inches between. Gently roll the sheets out to approximately 10"x10".
Brush the egg wash over the entire surface of each sheet.
Special note: Spread the filling over the pastry all the way out to 3 edges. Leave about 1/2 inch of pastry clear on the 4th edge to help seal the rolls closed.
Next, sprinkle with grated Parmigiano cheese, but save a little for after baking. Now, evenly distribute the prosciutto and then the arugula.
Starting with the edge opposite your sealing edge, roll each sheet into a cylinder. Transfer to a board or pan covered in parchment and pop into the fridge for 10 minutes.
Using a sharp knife, cut each roll into 8 equal pieces. Place the rolls, evenly spaced, into a lightly greased 9"x13" pan.
Bake the rolls for 20 to 24 minutes until golden brown and the pastry in the center of each roll is cooked through. Remove them to a cooling rack.
Brush melted butter over the tops and sprinkle with the last bit of Parmigiano-Reggiano. When ready to serve, stir a splash of water with the fig jam and microwave until it's a drizzling consistency. Serve on the side to drizzle over the rolls.