Cook edamame in large pot of boiling salted water until tender, about 6 minutes. Drain. Transfer to processor. Add both cheeses; process until smooth. Transfer to strainer set over large bowl. Press mixture through strainer with spatula (solids may remain in strainer). Season filling with salt. Cover and chill until firm, about 4 hours.
Line baking sheet with waxed paper. Place 1 goyza wrapper on work surface. Place 1 tablespoon filling on wrapper. Brush edges with glaze; fold over and seal. Transfer to sheet. Repeat with remaining wrappers, filling, and glaze.
Melt butter in large skillet over medium heat. Cook until brown, stirring often, about 5 minutes. Add mushrooms and sauté until soft, about 3 minutes. Season with salt and pepper.
Meanwhile, working in batches, cook ravioli in large pot of boiling salted water until ravioli rise to surface, about 3 minutes. Cook 1 minute longer. Using slotted spoon, drain ravioli well and transfer to skillet with sauce.
Toss ravioli gently over medium heat to coat and rewarm sauce. Divide ravioli and sauce among 6 shallow bowls, top with shaved parmesan cheese; serve.