Preheat oven to 350°. Coat a shallow 2 ½-quart baking dish with vegetable oil spray and set aside. Spread walnuts and Brazil nuts on large baking sheet or in a shallow pan. Bake, stirring once or twice, until lightly browned and fragrant, 10 to 12 minutes. Let cool slightly, then chop coarsely.
Place dried mushrooms in a small bowl. add warm water to cover. Let stand until softened, about 30 minutes. Remove mushrooms, squeezing out excess liquid. Chop mushrooms coarsely. Place a fine sieve over a 2-cup measuring cup. Strain the liquid into the cup.
Melt half the butter in a large frying pan over medium heat. Add onion and celery; cook, stirring occasionally, until softened but not browned, 5 to 7 minutes. Stir in parsley, thyme, and sage; cook 1 minute longer. Scrape mixture into a large bowl.
In the same frying pan, melt the remaining ¼ cup butter over medium heat. Add rehydrated and fresh mushrooms. Cook, stirring, until mushrooms are tender and lightly browned on the edges, 5 to 7 minutes. Stir in garlic, ½ teaspoon salt, and pepper; cook 1 minute longer. Add mixture to the onion and celery, then add walnuts, Brazil nuts, and bread pieces.
Add enough turkey stock to the mushroom liquid to equal 2 cups. Pour over the bread mixture, tossing gently to mix. Bread should be moist but not soggy; add stock as needed, depending on the dryness of the bread. Taste, adding more salt if needed. (If dressing will be used to stuff a turkey, cover and let cool. Don't stuff turkey until just before roasting.)
Spoon dressing into baking dish and cover with foil. Bake, covered, until heated through, about 30 minutes. If a crustsy topping is desired, remove foil and drizzle with turkey drippings. Continue baking, uncovered, until lightly browned on top, about 15 minutes longer.