Sort, rinse, and soak. Sort dried white beans, discard debris, and rinse. Cover with the soaking water in a large bowl and refrigerate 12 hours at 40°F (4°C) or colder, leaving room for expansion.
Drain and prepare. Discard soaking water; rinse and drain beans. Cut beef shoulder into 1 1/4-inch (3 cm) cubes. Dice onion to 1/4 inch (5 mm), slice carrots 3/8 inch (1 cm), and mince garlic. Keep beef cold; clean hands, utensils, and surfaces after raw-meat contact.
Brown the beef. Heat the main-pot oil in a heavy pot of at least 6 liters. Brown beef in uncrowded batches over medium-high heat, about 6–8 minutes total. Add pots or batches when scaling.
Soften the vegetables. Add onion and carrots and cook over medium heat about 5 minutes. Stir in garlic for 30 seconds without scorching.
Add beans and water. Add drained beans, bay leaves, and fresh main cooking water. Allow generous boiling headroom. Reserve extra water for later adjustments.
Boil for 30 minutes. Bring uncovered to a full boil, then maintain a definite boil for at least 30 minutes, timing from when boiling begins. Skim foam as needed. Keep beans surrounded by liquid, adding hot water as necessary. Do not replace this stage with gentle warming or a slow cooker, and do not taste beans during this stage.
Simmer until nearly tender. Reduce heat, partially cover, and simmer another 60 minutes. Stir carefully occasionally and check liquid. After the boil and simmer, check several beans. If centers are still firm or chalky, cook another 30–60 minutes or longer as needed. Do not start the roux until beans are nearly fully tender.
Make a light roux. Heat the separate roux oil in a small pan. Stir in flour over low to medium heat for 2–3 minutes until lightly golden, not dark.
Add paprika off the heat. Remove the pan from the burner. Stir sweet paprika into the warm roux for only a few seconds. Do not let it darken or burn.
Loosen the roux. Whisk in about 1 cup (250 ml) hot liquid taken from the bean pot per base batch, gradually, until smooth. This is borrowed liquid, not additional water.
Thicken and season. Stir the smooth roux into the stew. Add salt and black pepper. Simmer gently uncovered another 15 minutes, sweeping the bottom regularly to prevent sticking. Add reserved hot water in small amounts if needed.
Check beans and beef. Beans must be fully soft and creamy inside, with most retaining their shape. Beef should separate easily with a fork. This long-braised version uses a conservative center check of at least 160°F (71°C), but tenderness may take longer. The broth should coat a spoon but remain ladleable. Continue cooking if needed.
Rest and serve. Remove bay leaves, take the pot off heat, and rest 5 minutes. Stir, check seasoning, and serve one sixth of the batch per person. Bread is additional.
Cool and store promptly. Divide into shallow containers and refrigerate within 2 hours, or 1 hour above 90°F (32°C). Store at 40°F (4°C) or colder for 2–3 days. Freeze promptly chilled portions for longer storage. Thaw in the refrigerator and reheat throughout to 165°F (74°C), adding water as needed.