Sauté the shiitakes and bake the prosciutto - Preheat the oven to 375 ℉, convection bake setting. Destem and slice the mushrooms. Add them to a cold pan with a bit of salt and cook them, stirring often, until they've given up most of their water and the water has evaporated. Set aside. Lay the prosciutto slices on a lined baking sheet and bake until crispy, about 15 minutes. Let it cool, then crumble it and set it aside.
Make the scallion cream - Smash three garlic cloves. Place the chopped scallions, the smashed garlic, and the cream in a small saucepan. Bring mixture to a boil and then reduce the heat to low. Simmer, stirring occasionally until scallion greens are tender and the garlic is softened. Carefully process in a blender until the cream sauce is smooth and silky. Stir in 1/4 t salt and set aside.
Clean the shrimp and steam the asparagus - remove the shells and tail and devein if necessary. Remove the tough ends of the asparagus and peel them if necessary. Steam the asparagus until just crisp tender, then rinse under cold water to stop the cooking and preserve the color. Chop into 1" pieces, reserving the tips for garnish.
Cook the Pasta - Cook the pasta according to the package directions until 3 minutes shy of the stated cooking time. The pasta will finish cooking it in the sauce.
Assemble the dish - While the pasta is cooking, mince the remaining 3 cloves of garlic and cook in 2 T butter until fragrant. Add the scallion cream sauce, the shrimp, the asparagus, and the shiitakes to the pan and simmer until the shrimp is just beginning to become opaque. Then scoop the pasta into the pan, adding a 1/4 cup of the pasta water (or a bit more it it seems too thick). Cook until the the pasta is just tender, and the sauce is loose and creamy.
Finish, off the heat - Stir in the cheese, mint, lemon zest and juice, remaining 2 T of butter. Then pour into your serving bowl. Top with the crumbled, crispy prosciutto and serve.