In a small bowl, combine mushrooms and 2/3 cup boiling water. Let stand until soft, about 20 minutes. Rub mushrooms gently to release grit, then lift from water, squeeze dry, and finely chop. Reserve liquid.
Meanwhile, combine garlic, onions, and oil in an 8- to 10-quart pan over medium-high heat. Stir often until onions are lightly browned and taste sweet, about 10 minutes.
To pan, add mushrooms and most of the soaking liquid, discarding the gritty remainder. Also add tomatoes, parsley, wine, thyme, sugar, and chili flakes. Bring to a boil and simmer rapidly until reduced to 3 quarts, about 1 1/4 hours; stir often. Add salt to taste.
Use hot, or let sauce cool, then cover and chill up to 5 days. To store longer, freeze in easy-to-use portions.