In a large skillet heat butter and olive oil over medium-high heat until foamy. Stir in fennel, onions, and green onions. Sprinkle with salt and pepper. Cook and stir about 6 minutes or until vegetables are soft. Add mushrooms; cook and stir about 4 minutes or until mushrooms are slightly soft.
Transfer mixture to a large bowl. Stir in bread cubes, 3/4 cup of the Parmesan cheese, and the dried tomatoes. (To serve today, omit Step 3 and continue as directed in Step 4.)
Let mixture cool; transfer to a resealable plastic bag. Seal bag; chill for up to 2 days.
Preheat oven to 375 degrees F. Grease a 3-quart rectangular baking dish. Transfer bread mixture to prepared dish. Drizzle with broth. Carefully stir in tarragon, thyme, and basil. Bake, covered, for 40 minutes. Sprinkle with the remaining 1/4 cup Parmesan cheese. Bake, uncovered, for 20 to 25 minutes more or until hot in center (165 degrees F).