Using a food processor, finely chop half of the mushrooms, the roasted garlic, halved sun-dried tomatoes and chopped sage; stir in 1/2 teaspoon salt.
In a medium skillet, heat 1/4 cup olive oil over medium-high heat. Add the mushroom mixture and cook, stirring, for 3 minutes. Transfer to a bowl and let cool slightly; stir in 6 tablespoons cheese.
Meanwhile, heat 1/4 cup olive oil in the same skillet over medium-high heat; add the remaining mushrooms. Cook until golden, about 7 minutes. Remove from the heat, stir in the chopped sun-dried tomato and season with salt and pepper; let cool.
Bring a large pot of salted water to a boil; lower the heat and bring to a simmer. Lay 20 wonton wrappers on a work surface and place a bowl of water alongside. Place 1 tablespoon chopped mushroom mixture in the center of each wrapper. Using a pastry brush, dampen the edges with water; cover with another wrapper, pressing around the filling to push out the air and seal. Working in batches, add the ravioli to the simmering water and cook for 3 minutes; remove with a slotted spoon.
Divide the ravioli among 4 plates; top with the cooked mushrooms, prosciutto and remaining 2 tablespoons parmesan. Drizzle 1 tablespoon olive oil on each plate and scatter the sage strips on top.