Heat oven to 350 degrees. Place onions in a large bowl, cover them with boiling water, and let stand for 10 minutes. Drain onions in a colander. When cool enough to handle, peel onions, and transfer them to another bowl.
Add butter, fresh thyme, 2 teaspoons rosemary, and dried thyme to onions, and toss well. Season with salt and pepper.
Transfer onions to a large, shallow roasting pan, and arrange in a single layer. Roast onions, stirring occasionally, until they begin to soften and turn brown, about 1 hour. Remove from oven.
Meanwhile, soak mushrooms in 2 1/2 cups very hot water for 30 minutes. Lift mushrooms out of soaking liquid with a slotted spoon, and strain mushroom liquid through a sieve lined with several layers of cheesecloth (to remove grit from liquid). Chop strained mushrooms coarsely. Add strained liquid and chopped mushrooms to roasting pan with onions, and stir to combine.
Return pan to oven, and continue cooking, stirring occasionally, until liquid is reduced by three-quarters and onions are soft and brown, about 1 hour. Adjust seasoning; just before serving, sprinkle with remaining teaspoon rosemary.