For the puff pastry slices with San Daniele ham, roll out the puff pastry on a floured work surface to a thickness of about 0.5 cm. Brush with beaten egg yolk and sprinkle with freshly grated Parmesan. Preheat the oven to 220°C.
Cut the dough into pieces of about 7 x 10 cm and place them on a baking tray lined with baking paper. Chill again for 20 minutes. Place in the oven, turn the heat down to 180°C and bake the slices for about 10 minutes. Remove and let cool.
Meanwhile, peel figs and mash finely with a fork. Add olive oil and a few drops of balsamic vinegar and season with salt and pepper. Cut the cooled slices crosswise and spread one side with the fig puree. Top with some arugula as well as prosciutto and place the thinly shaved Parmesan on top.
Place the second half on top and serve the puff pastry slices with San Daniele ham.