Thoroughly pat the meat dry with paper towels and generously season it with salt and pepper.
In a Dutch oven over high heat, heat half of the oil until shimmering. Working in several batches, and without moving the meat much, sear the meat on all sides until well browned, adding more oil as needed. (If you try to cook too much meat at once, it will steam and turn gray instead of brown.) Once the meat is well browned, transfer to a plate.
Reduce the heat to medium-high and add the onions and any remaining oil to the pot. Cook, stirring from time to time, until the onions have softened and turn golden, about 10 minutes. Sprinkle the flour on top and cook, stirring occasionally, until thickened, 4 to 5 minutes. Add the wine and, using a wooden spoon, stir, scraping up all the browned bits (fond) off the bottom of the pot.
Once the wine starts to boil, return the meat and its accumulated juices to the pot, and add the carrots, garlic and the bouquet garni. Add 1 1/2 cups of water (and about 2 tablespoons of demi-glace, if you have it). Bring to a boil, then reduce the heat to medium-low and cook, uncovered, until the meat is tender, 2 to 2 1/2 hours, skimming off any foam or oil that might accumulate on the surface. Check on the stew every 15 to 20 minutes, stirring and scraping the bottom of the pot to prevent scorching or sticking.
As you check on the stew, continue adding 1/4 cup to 1/2 cup water, as needed, up to 2 1/2 to 3 cups total — to ensure there is enough liquid to cook down and concentrate. If the stew begins to stick, reduce the heat to low. The onions should fall apart, creating a thick, rich sauce that coats the meat.
When the stew is done, discard the bouquet garni, taste the stew and season with more salt, if desired. Garnish with the chopped parsley and serve.