Place the porcini mushrooms in a medium heat-safe bowl. Pour the boiling water over top and let the porcini soak until the liquid is a very dark brown, about 20 minutes. Strain the liquid into a small liquid measuring cup and save it (you should have about 3/4 cup), then chop the porcini and set aside.
Heat 2 tablespoons of the olive oil in a large heavy-bottomed pot over medium-high heat until shimmering. Add the cremini mushrooms, 1 tablespoon of Black Dog Belly Rub, and a little salt. Stir and try to keep the cremini in an even layer. Let them, untouched, until starting to caramelize, about 3 minutes. Stir a few times and then continue to cook, untouched, until golden brown, about 3 minutes more. Transfer to a medium bowl with a slotted spoon and return the pot to the burner.
Add another 2 tablespoons of olive oil to the pot along with the carrots and onion. Cook, stirring occasionally until the vegetables are soft, 4 to 5 minutes. Stir in the tomato paste, thyme, oregano, garlic and porcini mushrooms and cook, stirring frequently, until the tomato paste is toasted and turns a deep maroon color, 2 to 3 minutes.
Stir in the wine, bring to a simmer, and cook until reduced by half, 3 to 4 minutes. Fold in the pearl onions, cannellini, cremini and reserved porcini water and bring to a simmer. Cook, stirring occasionally until the sauce is thick enough to coat the back of the spoon and the mushrooms are very soft, 8 to 10 minutes. Fold in 3/4 cup of the parsley and use the rest of the parsley to garnish.Serve in a bowl with crusty french bread rubbed with garlic and olive oil.