In a large Dutch oven or heavy-bottomed pot, heat the 4 Tbsp of salted butter over medium heat until it melts.
Add the peeled whole pearl onions to the pot, and cook gently over medium heat, stirring carefully every few minutes.
Cook the onions until they are tender, or roughly 15 minutes.
*If your onions begin to take on too much color, add a small splash of water to the pot.
Carefully remove the onions from the butter using a slotted spoon, and place them in a bowl for later.
*Leave the remaining butter that you used to cook the onions inside the bottom of the heavy-bottomed pot.
Add the sliced mushrooms and cook, stirring occasionally, until they release their liquid and become tender, about 8-10 minutes.
Remove the onions from the butter using a slotted spoon and set them aside with the pearl onions that you cooked earlier.
Add the chopped carrots and onions to the pot and cook for an additional 7 minutes.
Add the chopped garlic, and cook for an additional one minute.
Stir in the tomato paste and flour and additional 2 Tbsp of salted butter, and cook for another 2-3 minutes.
Add the red wine and vegetable broth to the pot, and stir to combine.
Bring the mixture to a simmer, then reduce the heat to low.
Let the Mushroom Bourguignon simmer for about 45 minutes, or until the vegetables are tender and the sauce has thickened.
Add half of the cooked mushrooms from earlier, and continue to cook for an additional 30 minutes on low heat.
Taste, and season with Kosher salt and freshly-ground black pepper.
When you are ready to serve, add the sauce with the cooked vegetables and mushrooms to bowls, topped with the remaining cooked mushrooms and pearl onions from earlier.
Garnish each bowl with a sprinkling of fresh parsley.