Bring a medium-size pot of salted water to a boil. Set up an ice-water bath next to the stove.
Drop the fava beans into the boiling water and cook until very tender, 6 to 8 minutes. Drain the beans and immediately drop them into the ice bath to cool; drain them again and peel off their skins.
Heat the oil in a medium-size skillet over medium heat. Add the shallot and cook, stirring often, until it just begins to soften and become translucent, about 2 minutes. Add the fava beans and season them with a couple of pinches of salt and grinds of black pepper. Continue to cook, stirring occasionally, until the fava beans are completely soft, about 8 minutes. Add ¼ cup of water (if the beans start to stick to the bottom of the pan, add the water sooner) and continue to cook until the water is mostly absorbed, about 1 minute.
Transfer the mixture to a medium-size bowl and mash the fava beans with the back of a fork or a potato masher until they make a creamy but textured mash. Stir in 2 tablespoons of your best extra-virgin olive oil and half of the sliced mint, reserving the rest for garnish. The mash should be creamy and spreadable; if it seems dry, add up to 1 more tablespoon of olive oil and/or 1 tablespoon of water until it reaches the desired consistency. Adjust salt and pepper to taste. (If you are making the mash ahead of time, add the mint just before serving. The spread will keep, in an airtight container, for several days in the refrigerator; you may need to stir in a splash of water and olive oil if it dries out, as well as additional fresh mint.)
To serve, spread the mashed fava beans on crostini or crackers, or serve it in a bowl with the crostini alongside. Top with a slice of young Percorino or freshly grated aged Percorino, if you wish, and garnish with the remaining mint.