Basil Vinaigrette: Blend mustard, egg yolk, shallots and garlic in a food processor. With machine running, add oil slowly in a thin steady stream. Transfer mixture to a small bowl and whisk in lemon juice, vinegar and basil. Season with salt and pepper. Cover and refrigerate.
When ready to serve, cut asparagus diagonally into 1-inch pieces. Combine asparagus, mushrooms, onion, tomato and parsley. Toss with vinaigrette and season with salt and pepper. Mound asparagus mushroom salad into radicchio cups and place on a serving platter.