Combine water, a big pinch of salt, porcini mushrooms, and shiitake stems in a small saucepan. Bring to a simmer over medium heat, reduce heat, and gently simmer for about 20 minutes. Strain, discarding solids and reserving broth: you should have about 3 cups broth.
Wipe mushrooms with a damp towel to remove any dirt. Thickly slice shiitakes, quarter large crimini and white mushrooms, and halve smaller ones. Melt 1 tablespoon ghee in a large Dutch oven over medium heat. Working in batches, add mushrooms and cook, stirring occasionally, until browned in spots, about 5 minutes per batch, adding additional ghee as needed. Transfer mushrooms to a plate.
Reduce heat to medium-low and melt 1 tablespoon ghee in now empty pot. Add pearl onions. Cook, stirring occasionally, until browned all over, about 10 minutes. Transfer to plate with mushrooms.
Melt remaining 1 tablespoon ghee in pot and add chopped onions and carrots. Season with salt and cook until onions are soft and sweet, 5–10 minutes. Add tomato paste and garlic and cook for 1 minute, stirring continuously. Add cognac and use a wooden spoon to scrape brown bits from bottom of pan. Add wine and bring to a simmer. Add reserved porcini broth, thyme, bay leaf, and return mushrooms and pearl onions to pot. Bring to a gentle simmer and cook, uncovered, for 20 minutes.
Combine butter and flour in a small bowl to make a paste. Gradually whisk into stew. Season with salt and pepper and cook for an additional 20 minutes. Sprinkle with parsley and serve with hunks of bread and roasted potatoes, if desired.
NOTE: Ghee is clarified butter and can be found in health food stores and gourmet grocers; substitute with vegetable oil if necessary. To peel pearl onions, bring a large saucepan of water to boil. Drop in onions and blanch for about 1 minute. Drain and let cool enough to handle. Use a sharp knife to slice the root end off and pop each onion out of its skin.