Combine first 5 ingredients in a large roasting pan. Roast at 450° for 45 minutes or until liquid evaporates; stir twice.
Meanwhile, combine vinegar and next 5 ingredients in a small bowl with a wire whisk; slowly whisk in 2/3 cup oil.
Combine mushrooms and 1/4 cup vinaigrette; cool to room temperature. Divide spinach and radicchio among serving plates; top with mushrooms and cheese. Drizzle with remaining dressing. Serve immediately.
Make Ahead: Chill the vinaigrette in a jar up to 3 days; whisk well before serving.