Heat one tablespoon olive oil and one tablespoon butter in a medium Dutch oven or heavy saucepan over high heat. Sear the mushrooms and pearl onions in batches until they begin to brown slightly, but the mushrooms have not yet released any liquid — about three or four minutes. Remove them from the pan and set aside.
Reduce heat to medium and add another tablespoon of olive oil. Add the carrots, onions, thyme, a few good pinches of salt and several grinds of black pepper into the pan and cook for 10 minutes, stirring occasionally until the onions are lightly browned. Add the garlic and cook for one more minute, until fragrant.
Add the wine to the pot, scraping any stuck bits off the bottom. Turn heat to high and boil until reduced by half. Stir in the tomato paste and the broth. Add back the mushrooms and pearl onions with any juices that have collected. Once it boils, reduce to a simmer for 20 minutes, or until mushrooms are very tender.
Combine remaining butter and the flour with a fork until combined. Drop it into the stew by small pieces, then stir well. Reduce heat and simmer 10 minutes longer. If the sauce is too thin, boil it down to reduce to the right consistency. Season to taste.
To serve, spoon the stew over a bowl of egg noodles, dollop with sour cream and sprinkle with chives or parsley (optional).