Can be prepared in 45 minutes or less.
In a kettle of boiling salted water cook pasta until al dente.
While pasta is cooking, in a large non-stick skillet heat 1 tablespoon oil over moderately high heat until hot but not smoking and sauté all mushrooms with salt and pepper to taste, stirring occasionally, until golden brown.Add garlic and 1 tablespoon vinegar and cook, stirring, 1 minute. Add broth and peas and simmer 3 minutes.
Drain pasta in a colander and stir into mushroom mixture. Toss mixture and remove skillet from heat. In a large bowl whisk together mustard, remaining tablespoon oil, and remaining 2 tablespoons vinegar until blended. Add radicchio and toss to coat with dressing. Add pasta mixture and Parmesan and toss well.