Preheat oven to 375 degrees F. Grease a 3-quart rectangular baking dish; set aside. In a large skillet heat butter and oil over medium-high heat until foamy. Add yellow onions, fennel, green onions, salt, and pepper. Cook and stir about 6 minutes or until vegetables are tender. Add mushrooms; cook and stir about 4 minutes or until mushrooms are slightly softened. Transfer mixture to a large bowl.
Stir bread cubes, 3/4 cup of the cheese, and the dried tomatoes into vegetable mixture. Transfer mixture to the prepared baking dish. Drizzle with broth to moisten, tossing gently to combine. Gently stir in tarragon, thyme, and basil.
Bake, covered, for 40 minutes. Sprinkle with the remaining 1/4 cup cheese. Bake, uncovered, for 20 to 25 minutes more or until heated through.