In a saucepan, melt 3 tablespoons of the butter. Add the onion and cook over moderate heat until softened, 7 minutes. Add the red wine, beef stock and thyme and bring to a boil. Simmer until the liquid has reduced to 3/4 cup, 1 hour and 15 minutes; strain into a clean saucepan. Season with salt and pepper, cover and keep hot.
Meanwhile, bring a medium saucepan of water to a boil. Add the wild rice and a pinch of salt, cover and simmer over low heat until tender, about 25 minutes. Drain the rice and return it to the saucepan. Season with salt, cover and keep warm.
Preheat the oven to 325°. In a medium skillet, melt the remaining 2 tablespoons of butter. Add the shiitake and season with salt and pepper. Cover and cook, stirring occasionally, until tender and browned, about 5 minutes.
Arrange the salmon fillets on a rimmed baking sheet, skinned side down, and season with salt and pepper. Bake for about 15 minutes, until just cooked in the center.
Spoon the wild rice onto plates and set the salmon fillets on top. Spoon the shiitake mushrooms and red wine sauce over the fish and serve.