Melt 5 tablespoons butter in heavy large skillet over medium-low heat. Add onions and sauté until brown, 40 minutes.
Meanwhile, cook rutabagas in large pot of boiling salted water until tender, about 20 minutes. Drain well.
Melt 3 tablespoons butter in large skillet over medium-low heat. Add rutabagas; sauté until heated through, about 10 minutes. Drizzle honey over. Gently stir in onions. Season with salt and pepper. (Can be made 3 hours ahead. Let stand at room temperature. Rewarm over medium-low heat.)