Preheat oven to 450°. Place onions and carrots each on a separate baking sheet with sides. To each sheet, add 8 thyme sprigs, 1 tablespoon olive oil, 1/2 teaspoon salt and 1/4 teaspoon pepper; toss until vegetables are coated evenly. Roast 25 minutes, tossing vegetables several times until lightly caramelized and slightly tender; cool.
Quarter cremini mushrooms; remove stems from shiitake mushrooms and slice caps; slice oyster mushrooms. In a large, deep skillet over high heat, melt butter and sauté mushrooms until tender, about 4 minutes. Add roasted onions, carrots, broth and vinegar. Cook, tossing vegetables with a large spoon, until broth and vinegar have evaporated and vegetables are heated through, 3 to 4 minutes. Add tarragon and remaining 1/4 teaspoon salt and 1/4 teaspoon pepper; toss.