Heat 2 tablespoons of the olive oil in a large heavy-bottomed pot over medium-high heat until shimmering. Add the white and baby bella mushrooms, and stir so they are in an even layer. Let the mushrooms cook, untouched, until starting to caramelize, about 3 to 5 minutes. Stir them a few times and then continue to cook, untouched, until golden brown, about 3 or 4 more minutes. 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 softened, 4 to 5 minutes. Stir in the tomato paste, thyme, oregano, and garlic. Cook, stirring frequently, until the tomato paste turns a deep maroon color, 2 to 3 minutes.
Stir in the wine, and bring to a simmer. Cook until reduced by half, 3 to 4 minutes. Fold in the pearl onions, cannellini, reserved 3/4 cup vegetable stock, Black Dog Belly Rub, 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 1/2 cup of the parsley.
While the bean and mushroom mixture is cooking, mix minced garlic with 4 tablespoons of olive oil and brush both sides of the French bread. Please one a cookie sheet and broil until golden brown on both sides. Serve with the bourguignon.