Place ham bone in a large pot or Dutch oven and cover with 6 cups water (or broth). Bring to a boil, then reduce heat and simmer uncovered for 30–45 minutes. Remove the bone and set aside to cool. Discard bay leaf if added early.

In a separate skillet or the same pot (after removing the bone), heat olive oil or butter over medium heat. Add onions, garlic, carrots, and celery. Cook until softened, about 5–7 minutes.

Pour the reserved ham broth back into the pot (if you removed it) along with the cooked vegetables. Add potatoes, bay leaf, thyme, and pepper. Bring to a boil, then reduce to a simmer and cook for 20–25 minutes, or until potatoes are tender.

Remove meat from the ham bone and shred or chop into bite-sized pieces (discard fat or gristle). Stir the ham back into the pot.

In a small bowl, whisk flour with milk or half-and-half until smooth. Stir this into the soup and simmer for 5–10 more minutes until slightly thickened. (Or mash a few potatoes in the pot for natural thickening.)

Add salt only if needed (ham is usually salty). Season with more pepper if desired. Remove bay leaf before serving.

Serve hot with crusty bread or rolls.