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1
Preheat the oven to 275F (135C).
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2
Heat a heavy bottomed, oven-safe pot over medium-high heat (I use an enamel cast iron dutch oven for this). Swirl about a tablespoon of olive oil into the pot. Add the onion, garlic, and carrots; cook 2 to 3 minutes, until the onions are lightly golden. Use a slotted spoon to transfer the vegetables to a bowl.
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3
Generously season the meat with salt and pepper. Don't be shy with the salt, as it enhances the overall flavor. Raise the heat to high, and let the pot get very hot, so it'll be ready to brown the beef properly. Swirl a bit more oil into the pot, just enough to cover the bottom, if necessary. Add the meat, and cook until deeply browned all over. Transfer the beef to the bowl with the vegetables.
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4
Pour in the vermouth, and use a wooden spoon to scrape up any browned bits from the bottom of the pot. Bring to a boil, then reduce to simmer. Let cook 2 to 3 minutes until the alcohol has dissipated, and reduced by 1/3. Whisk in the butter until it's completely dissolved. Sprinkle in the flour, and whisk until the mixture is smooth (this is called making a roux, in case you ever wondered what that term means). Cook, whisking constantly, 1 to 2 minutes, until the color deepens a bit (this cooks the flour so it doesn't add a raw taste to your sauce). Slowly pour in the stock, whisking constantly, until it becomes a slightly thickened gravy.
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5
Add the meat back to the pot. Spoon the vegetables on top, and add the herbs. Cover with a tight fitting-lid (or foil if you don't have a lid). Place the pot in the oven, and cook for 4 hours. Remove the lid, and cook for 30 minutes more, uncovered, until the meat is falling apart-tender when pierced with a fork. While it's a lovely meal on its own, I like to serve pot roast with mashed potatoes.
