In a 4- to 6-quart pan over high heat, bring 2 quarts water to a boil. Add pasta and cook until tender to bite, about 10 minutes. Drain, rinse with cold water, and drain again.
Meanwhile, cut common and shiitake mushrooms into irregular chunks. In a 12-inch frying pan or 14-inch wok over medium-high heat, melt 1 tablespoon butter with the olive oil. When butter is foamy, add mushrooms. Sprinkle with salt and pepper and cook, stirring often, until mushrooms are lightly browned, 4 to 5 minutes.
Add remaining tablespoon butter to pan. When melted, add garlic and ginger and stir until fragrant, 1 minute. Add radicchio and edamame and stir until radicchio is wilted, 1 to 2 minutes. Remove from heat.
In a large bowl, whisk vinegar, mustard, and soy sauce until combined. Add pasta and mushroom mixture and stir gently to coat. Add salt and pepper to taste. Serve warm or let cool to room temperature; top with shaved parmesan cheese just before serving.