Preheat oven to 375 degrees Fahrenheit. Bake the prosciutto slices in the oven until crisp. Set aside. On an unlined baking sheet, bake the potatoes, skin on, for one hour, or until potatoes are soft. While the potatoes are baking, begin steeping the dried mushrooms. In a sauce pot with lid, combine 2 cups broth and 1 cup wine, and allow to come to full boil over medium-high heat. Add dried mushrooms and completely submerge, turning heat to a medium-low. Allow to cook for 30 minutes. After 30 minutes, remove from heat, cover pot with lid, and steep for one hour. Strain the liquid from the mushrooms and reserve in a bowl. Roughly chop the mushrooms and set aside.
Pull the softened potatoes from the oven, allow to cool slightly, and peel. Pass through a ricer or a potato mill over a large bowl. Add the eggs, cheese, salt and pepper. Mix with hands until incorporated, allow to completely cool. Once cooled, add the flour, using your hands, and lightly mix. Prepare a lined baking sheet with a dusting of flour, set aside. Working on a lightly floured surface, divide the dough into four equal pieces. Begin rolling into 1 1/2 inch-thick ropes. Cut the ropes into three inch pieces, and place on prepared baking sheet. Once completed, place parchment or plastic wrap over the baking sheet and place in fridge for 30 minutes or up to one hour. While gnocchi rest, place a stockpot filled with water and seasoned with salt on the stove. Allow to boil.
Place a saute pan over mediumhigh heat. Add 1/4 cup olive oil and 1 Tbsp butter and allow to heat until shimmery. Add the cremini mushrooms, salt, pepper, chopped thyme, and saute until slightly browned and fragrant. Place the mushrooms in a bowl and set aside. With saute pan on medium-high, add remaining olive oil & butter. Once shimmering, add the garlic, leeks, and season with salt and pepper. Saute leeks until reduced by three-quarters. Once reduced, combine both the dried & sauteed mushrooms with the leeks. Add 1 cup of mushroom broth. Turn heat to medium, and allow the liquid to reduce, stirring occasionally. Once the broth has reduced by half, turn heat to medium-low and add remaining broth, stirring occasionally until reduced to a thick consistency.
While mushroom ragu is thickening, begin cooking the gnocchi. Prepare a saute pan with 2 Tbsp of oil over medium heat. Start dropping gnocchi, a handful at a time, into the boiling water and cook until gnocchi float to the surface. Allow to float for 30 seconds then pull from pot with a slotted spoon. Shake off any excess water then immediately add to prepared saute pan. Allow gnocchi to slightly brown and crisp on all sides, then transfer to mushroom ragu. Repeat process until all gnocchi has been cooked. Once plated, garnish with prosciutto crisps and a hefty portion of freshly grated cheese.