In a 12-inch, heavy nonstick skillet (enameled cast iron if you've got it), heat one tablespoon each of the olive oil and butter over high heat.
Add half the sliced mushrooms and cook without disturbing until golden brown on one side, about 3 minutes. Toss once and cook until some mushrooms start to brown on the other side (don't stress at all about flipping all of them perfectly), about two minutes more. (See note 6 below.)
Scrape into a large bowl and repeat with a second tablespoon each of olive oil and butter and the other half of the mushrooms.
Set mushrooms aside.
In the same skillet, warm the remaining tablespoon olive oil over medium-high heat.
Add onion and carrot along with the salt and pepper and cook, stirring occasionally, until tender and lightly browned, about 10 minutes.
Stir in the garlic, thyme (stems and all), and sage and cook until very fragrant, a minute or two.
Pour in the wine. Turn heat up to high and simmer briskly until reduced by half.
Stir in tomato paste and then broth.
Add back the mushrooms and all the beautiful accumulated juice from the bowl. Bring to a boil, then simmer for 20 minutes. (This is a good time to start the mashed potatoes.)
With a fork, thoroughly mash together the remaining tablespoon of butter and the flour in a small bowl to form a paste.
Stir paste into stew.
Lower heat to simmer gently and cook 10 minutes more. Sauce should be thick and glossy.
To serve, remove thyme stems. Serve stew spooned over mashed potatoes.