Preparation Boil onions in large saucepan of boiling salted water 3 minutes to loosen skin. Drain. Peel onions. Bring chicken and beef broths to boil in large pot. Add turnips; reduce heat to medium, cover, and simmer until just tender, about 10 minutes. Drain, reserving 3 cups broth. Melt butter in same pot over medium-low heat. Add flour and thyme; stir 1 minute. Whisk 3 cups broth into pan, increase heat to medium-high and whisk until sauce thickens slightly, about 2 minutes. Add onions, reduce heat to medium, and simmer until tender, about 12 minutes. Stir in cream and turnips. Simmer until vegetables are coated with sauce, about 2 minutes. Season to taste with salt and pepper. (Can be made 1 day ahead. Cool slightly, cover and refrigerate in pan. Rewarm over low heat.) Preheat broiler. Transfer mixture to 11x7x2-inch glass baking dish. Sprinkle gratin with cheese. Broil until cheese melts and browns in spots and vegetables are heated through, about 5 minutes.
Preparation Boil onions in large saucepan of boiling salted water 3 minutes to loosen skin. Drain. Peel onions. Bring chicken and beef broths to boil in large pot. Add turnips; reduce heat to medium, cover, and simmer until just tender, about 10 minutes. Drain, reserving 3 cups broth. Melt butter in same pot over medium-low heat. Add flour and thyme; stir 1 minute. Whisk 3 cups broth into pan, increase heat to medium-high and whisk until sauce thickens slightly, about 2 minutes. Add onions, reduce heat to medium, and simmer until tender, about 12 minutes. Stir in cream and turnips. Simmer until vegetables are coated with sauce, about 2 minutes. Season to taste with salt and pepper. (Can be made 1 day ahead. Cool slightly, cover and refrigerate in pan. Rewarm over low heat.) Preheat broiler. Transfer mixture to 11x7x2-inch glass baking dish. Sprinkle gratin with cheese. Broil until cheese melts and browns in spots and vegetables are heated through, about 5 minutes.