In a medium saucepan, bring 1 cup broth, the wild rice and 1/2 cup water to a boil. Lower the heat, cover and simmer until about half of the grains pop, about 45 minutes. Drain and let cool.
Meanwhile, in a large, heavy skillet, melt 3 tbsp. butter over medium-high heat. Add half of the mushrooms and cook, stirring a couple of times, until nicely browned, about 7 minutes. Season lightly with salt and pepper and transfer to a plate. Repeat with the remaining mushrooms and 3 more tbsp. butter. Set all of the mushrooms aside in the skillet.
In a small saucepan, melt the remaining 2 tbsp. butter over medium heat. Add the onion, 1/2 tsp. salt and 1/4 tsp. pepper and cook, stirring occasionally, until softened, 6 to 7 minutes. Add the white rice and stir for 1 minute. Pour in the remaining 2 cups broth and bring to a boil over high heat. Cover, lower the heat and simmer until the rice is tender, 17 to 20 minutes. Fluff the rice with a fork and cover.
Add the edamame to the skillet with the mushrooms. Stir in the wild rice and cook gently over medium heat until heated through, 3 to 5 minutes. Transfer the wild rice mixture to a large bowl; add the white rice and thyme and toss well.