Pick through the beans to remove any stones or debris. Place them in a large bowl or pot, and cover with tap water by at least 2". Leave to soak overnight (or at least 8 hours, if you can start early in the morning).

Drain the beans and discard the liquid.

In a large 5.5 to 7 quart soup pot or Dutch oven, heat the oil over medium until the surface shimmers.

Add the kielbasa and saute for about 5 minutes until the edges start to turn golden (kielbasa is a fully cooked product; we're drawing out its flavors by browning it first).

Add the onions, carrots, celery, and garlic to the pot, and saute until softened (they'll also reduce a bit in volume), about 5 minutes.

Clear a space in the center of the pot and position the ham hock. Turn it, if necessary, so that the hock is sitting below the level of the pot (so you can cover it with a lid).

Add the drained beans to the pot.

Pour in 6 cups of water, covering the beans completely, plus the bay leaves. Bring the soup up to a bubbling boil, and then reduce the heat to maintain an active simmer. Skim off any foam that initially bubbles up from the ham or the beans.

If the soup is hissing water onto the stove, reduce the heat a little. Partially cover the pot with its lid or a piece of foil. Do not fully cover.

Start the clock — the soup will cook for a total of 2 hours from this point.

Check the soup every half hour if you can, and add more water to keep the beans covered. You might need to add water every time you check; you might not need to add water at all. It simply depends on the rate of evaporation as the soup cooks. If the water is boiling aggressively, reduce the heat a bit so that it's bubbling, not roiling.

Also give the soup a good stir from the bottom each time you check the pot.

When there are 30 minutes left in the cook, reduce the heat so that the soup calms to a gentle simmer, and fish out the bay leaves.

Add the tomatoes, mustard, chili powder, and W-sauce, stirring well. Reposition the lid and continue to the 2-hour mark.

At the end of the cook, turn off the heat and remove the lid.

Carefully lift the ham hock from the pot — use tongs and a spatula — and place it in a large bowl to cool down for several minutes.

Test the beans for doneness by smashing a few different types against the side of the pot. They should mush easily.

When the ham hock can be handled safely, cut any remaining meat from the bone, chop it up, and add it to the soup (meat may have fallen off the bone into the soup during the cook)

Carefully taste the soup — it's hot! Add salt to taste, if needed.