Place a rack in middle of oven; preheat to 350°. Sprinkle one 4-lb. boneless beef chuck roast all over with 2 Tbsp. Diamond Crystal or 1 Tbsp. plus ½ tsp. Morton kosher salt and 2 tsp. freshly ground pepper. Heat ¼ cup vegetable oil in a large ovenproof Dutch oven or other heavy pot over medium-high. Cook roast, turning occasionally, until browned all over, 15–20 minutes. Transfer to a large plate.

Reduce heat to medium and cook 2 small onions, finely chopped, and 2 small celery stalks, finely chopped, in same pot, stirring often and scraping up browned bits stuck to bottom, until golden brown and softened, 9–13 minutes. Add 10 garlic cloves, finely chopped, and one 3" piece ginger, peeled, finely chopped, and cook, stirring often, until fragrant, 2–3 minutes. Add 2 Tbsp. all-purpose flour and cook, stirring constantly, until combined, about 2 minutes. Gradually pour in ⅔ cup dark rum, stirring constantly to prevent clumping, then add 2½ cups low-sodium beef broth, 2 cups ginger beer, 3 Tbsp. fresh lime juice, 3 Tbsp. Worcestershire sauce, and 1 Tbsp. light brown sugar. Increase heat to high and bring to a boil. Cook, stirring occasionally, until sauce is reduced by about one third and thick enough to lightly coat a spoon (it should be the consistency of heavy cream), 15–20 minutes. Taste sauce and season with more salt and pepper if needed.

Remove pot from heat and place roast in center of pot. Arrange 1 lb. carrots, peeled, cut into 1" pieces, and 1 lb. small yellow potatoes (15–20), scrubbed, around meat, cover, and braise in oven until meat is tender enough to shred in the center without much resistance, 3–3½ hours. Uncover and turn meat over. Roast until meat is browned on top and sauce thickens slightly, 30–45 minutes. (If roast is browning too quickly, turn over and continue roasting until sauce thickens.) Skim as much fat from surface as possible; discard. Top with mint and/or cilantro leaves with tender stems (if using) and serve with lime wedges for squeezing over if desired. Do ahead: Roast can be cooked 3 days ahead. Let cool; cover and chill. Skim fat from surface before reheating, covered, in a 350° oven, 20–30 minutes.