Preheat the oven to 160°C fan.
Season the beef cheeks generously with salt and pepper.
Place the seasoned beef cheeks, onion, garlic, tomato paste, red wine, beef stock, thyme and bay leaf into a large ovenproof casserole dish. Stir the liquid to combine, cover with a lid and transfer directly to the oven. Place the pot towards the top of the oven with a rack free below to roast the carrots on later.
Braise for 3–3½ hours, turning the cheeks once or twice, until they’re extremely tender and easily pull apart with a fork. You'll add the carrots for the last hour of the beef cooking.
Toss the carrots with 1 teaspoon olive oil and a generous pinch of salt. Spread them over a lined tray and add to the lower rack of the oven that the beef is in, roast until completely soft and slightly caramelised around the edges.
Transfer the roasted carrots to a blender. Add 1 tablespoon of stock, the vinegar and a pinch of salt, then blend until completely smooth. Add another 1 -- 2 tablespoons of stock if needed to create a silky purée. You want to keep it as thick as possible.
Once the beef cheeks are ready, carefully lift them from the braising liquid and let rest for 10 minutes.
Simmer the remaining sauce/cooking liquid from the beef over medium-high heat until reduced, glossy and concentrated. If it needs thickening, mix together the flour and cold water and gradually whisk into the sauce until thickened and you have enough sauce to cover all three serves. Taste and adjust the seasoning.
Steam or blanch the beans until just tender, then season with salt.
Divide the carrot purée among three plates or containers and add the green beans and beef cheeks. Spoon the reduced red wine sauce generously over the top. Top with fresh parsley and enjoy!