Melt 2 tablespoons butter in a 6-quart pot. Add mushrooms and onion. Cook over medium-high heat 5 minutes. Add brown and wild rice only. Continue to cook, stirring often, 2 minutes or until bottom of pot starts to brown.
Stir in broth and white cooking wine. Increase heat to high; cover and bring to a boil. Reduce heat to low. Simmer, covered, 30 minutes.
Stir in white rice and carrots. Cover and continue to simmer 20 minutes more or until liquid is absorbed. Stir in thyme and remaining 2 tablespoons butter. Serve immediately or keep warm until ready to serve.