In a large bowl, combine porcini and 1 cup boiling water. Let stand until mushrooms are limp, about 20 minutes. Squeeze mushrooms gently to remove grit, then lift from water, squeeze dry, and finely chop.
Without disturbing grit, pour most of soaking liquid into a glass measure; add water to make 2/3 cup. Discard gritty liquid. Rinse and dry bowl.
Meanwhile, rinse quinoa in a fine strainer under cool running water.
Put quinoa in a 4- to 5-quart pan. Add the 2/3 cup porcini soaking liquid and the broth. Bring to boiling over high heat, then reduce heat and simmer, covered, until quinoa is just tender to bite, 10 to 12 minutes. Drain and save liquid, adding enough water to make 1/2 cup. Put quinoa in the large bowl.
As quinoa cooks, rinse common mushrooms, trim off and discard discolored stem ends, and finely chop.
In a 10- to 12-inch nonstick frying pan over medium-high heat, stir shallots and garlic until shallots are limp and tinged brown, 3 to 4 minutes.
Add porcini, common mushrooms, celery, and rosemary. Stir often over high heat until mushrooms are limp and juices cook away, about 5 minutes.
Mix vegetables with quinoa, bread pieces, parsley, thyme, and salt and pepper to taste.
Pour quinoa dressing into a shallow 2 1/2- to 3 1/2-quart casserole and add reserved cooking liquid. Cover and bake in a 400° oven until hot, about 25 minutes (30 minutes if chilled).