Melt 3 tablespoons butter in large nonstick skillet over medium-high heat. Add shiitake mushrooms and sauté until tender, about 5 minutes. Transfer mushrooms to medium bowl. Melt remaining 3 tablespoons butter in same skillet. Add shallots and garlic and sauté until tender, about 2 minutes. Add green beans and toss to coat with butter. Pour broth over green bean mixture. Cover and simmer until liquid evaporates and green beans are crisp-tender, about 10 minutes. Stir in shiitake mushrooms. Season to taste with salt and pepper. Transfer to platter and serve.