Place a large cast iron pot or skillet over medium heat and add oil.
When the oil is heated, add the onion and vegan meat. Cook, stirring occasionally, for approximately 15 minutes or until the onion is softened and the vegan meat is slightly crispy.
Add the garlic and cook for 1 minute until fragrant.
Add the vinegar and scrape the browned bits from the bottom of the pot.
Tear or rough chop the turnip greens and add them to the pot. Stir to combine.
Immediately add the salt and black pepper and stir to combine. Allow the turnip greens to wilt down to half of their original size (after approximately 2 - 3 minutes of cooking).
Add the brown sugar and broth. Stir to combine.
Bring the liquid in the pot to a simmer then cover with a lid and reduce the heat to low.
Cook for 45 minutes, stirring a few times while cooking. Make sure to keep the lid on the pot after stirring. Once done, taste the turnip greens and add any additional salt, black pepper, or brown sugar if preferred. Serve warm. Store the cooled leftovers in a food-safe airtight container in the fridge for up to 3 days.