Heat the olive oil and 1 Tbsp of the butter in a wide heavy pan over high heat. Add the mushrooms in batches so they brown instead of steaming. Let them sear for 2–3 minutes per side until deeply golden. Season lightly with salt. Transfer to a bowl and set aside.
In the same pan, add the carrot and sliced onion. Sauté until lightly browned. Return the seared mushrooms to the pot.
Sprinkle the salt, pepper, and flour over the vegetables. Toss to coat.
Place the uncovered pot in a preheated 450°F oven for 4 minutes.
Remove, toss again, and return to the oven for another 4 minutes.
Reduce oven temperature to 325°F.
Stir in the wine, 1 cup vegetable stock, tomato paste, garlic, thyme, and bay leaf.
Bring to a simmer on the stove. Cover and place in the lower third of the 325°F oven for 45–60 minutes, until the mushrooms are tender and the flavors have deepened. Add more stock as needed if the sauce reduces too far.
While the stew cooks, sauté the pearl onions until tender.
In a separate pan, sauté the 1 lb quartered mushrooms in butter until golden.
Set aside.
When the stew is done, pour the contents through a sieve into a saucepan, reserving the mushroom–vegetable mixture.
Simmer the sauce for 1–2 minutes, skimming any fat. Reduce until you have about 2½ cups of glossy, spoon-coating sauce.
Season to taste.
Return the mushroom mixture to the pot. Pour the finished sauce over it. Top with the pearl onions and sautéed mushrooms.
Simmer for 2–3 minutes, gently basting with the sauce.
Serve warm with mashed potatoes, buttered noodles, rice, or crusty bread.
Garnish with parsley.