Butter a shallow 13x9" baking dish. Bring a large saucepan of water to boil. Add the wild rice, cover partially and simmer over moderately low heat until tender, about 45 minutes. Drain well.
Meanwhile, melt 2 tablespoons butter in a medium saucepan. Add the chopped onions, garlic, and celery and cook over mmoderately high heat, stirring occasionally, until softened, about 6 minutes. Add the jasmine rice and cook for 2 minutes, stirring. Add 2 1/2 cups of water and 1 teaspoons of salt and bring to a boil. Cover and simmer over low heat until the water is absorbed and the rice is tender, about 18 minutes. Fluff the jasmine rise with a fork and transfer to a large bowl.
In a medium skillet, melt the remaining 2 tablespoons of butter. Add the sliced onions and cook over moderate heat, stirring occasionally, until softened and golden, about 15 minutes; add a few tablespoons of water if the onions dry out. Add the onions to the jasmine rice.
Add the sausage to the skillet and cook over moderately high heat, breaking it up, until browned and cooked through, about 8 minutes. Stir in the chives, sage, and thyme and scrape the mixture into the bowl. Add the chestnuts and drained wild rice, season with salt and pepper and stir gently.
Add the broth to the skillet and scrape up any browned bits stuck to the bottom of the pan. Boil until the broth is reduced to 1 1/2 cups, about 10 minutes. Make a slurry from 2 tablespoons cornstarch and 2 tablespoons water and add to the reduced broth. Cook, stirring, until thickened, 2-3 minutes. Stir the broth into the rice stuffin and spread the stuffing in the prepared dish. Cover with foil and bake until sizzling, about 30 minutes. Serve hot.
Unbaked stuffing can be refrigerated overnight. Bring to room temperature before baking.