Blot the chuck roast thoroughly with paper towels to remove excess surface moisture. Season every side evenly with ½ teaspoon fine sea salt and ½ teaspoon freshly ground black pepper.
Warm 1 tablespoon extra-virgin olive oil in a large oven-safe Dutch oven or sturdy, heavy-bottomed pot over medium-high heat until the oil is shimmering.
Place the seasoned beef in the hot pot and leave it undisturbed for 5–7 minutes per side. Continue searing until the exterior develops a pronounced golden-brown crust.
Transfer the browned roast to a plate and reserve it while preparing the onions.
While the beef is browning, slice the yellow onions into rings approximately ¼ inch thick.
Add the remaining 2 tablespoons extra-virgin olive oil to the same pot. Add the sliced onions and season them with ½ teaspoon each of fine sea salt and freshly ground black pepper.
Cook the onions over medium heat for approximately 5 minutes. Lower the heat to medium-low and continue cooking for another 20–25 minutes, stirring from time to time, until the onions are exceptionally soft and beginning to develop a golden color.
Allow the onions sufficient time to soften and caramelize, as this process provides the characteristic French onion flavor of the finished dish.
Stir the minced garlic and maple syrup into the softened onions. Continue cooking for 7–8 minutes, allowing the mixture to become fragrant and deepen slightly in color.
Scatter the all-purpose flour evenly over the onion mixture. Stir continuously for about 1 minute so the flour thoroughly coats the onions.
Pour in the red wine and use a wooden spoon to loosen the browned bits attached to the bottom of the pot, incorporating them into the sauce.
Add the beef stock, rosemary, and bay leaf. Stir gently and bring the liquid to a mild simmer.
Return the seared chuck roast to the Dutch oven, positioning it securely among the onions and braising liquid. The beef should be mostly submerged in the liquid.
Cover the pot tightly with its lid and place it in an oven preheated to 300°F (150°C). Braise for 3 hours.
After the initial 3 hours, carefully remove the pot from the oven and turn the roast over. If the meat has softened sufficiently, use two forks to gently begin separating it into larger pieces.
Cover the pot once more and return it to the oven for an additional 30–45 minutes. Continue braising until the beef is exceptionally tender and pulls apart readily with a fork.
Remove the finished pot roast from the oven and discard both the rosemary sprig and bay leaf.
Allow the beef to rest in its onion-rich braising sauce for approximately 10–15 minutes. This brief resting period gives the meat time to absorb some of the flavorful juices.
Shred or portion the tender beef as desired and serve generously covered with the rich French onion sauce and caramelized onions.