Combine water, ham hock, and pork shank in a large pot. Bring to a boil, reduce heat, and simmer until meat is cooked through, about 2 hours. Drain and set meat aside to cool to room temperature, about 20 minutes.

Remove meat from pork shank, chop into bite-sized pieces, and place in a bowl. Remove skin and meat from ham hock, chop into very small pieces, and add to the bowl, omitting skin if you prefer.

Heat bacon drippings in a large skillet or Dutch oven over medium-high heat until hot, but not smoking. Add 1/4 of the cabbage to the hot skillet and saute until slightly wilted, about 3 minutes. Add 1/4 of the meat mixture. Add another 1/4 of the cabbage, cooking until slightly wilted, about 3 minutes, followed by another 1/4 of the meat mixture. Repeat until all cabbage and meat has been added. Season with salt and pepper. If mixture sticks to the skillet, add 1 to 2 tablespoons chicken broth.

Slowly add remaining chicken broth to the cabbage-meat mixture in the skillet, give it a good stir, and reduce heat to medium-low. Simmer until cabbage is tender, 30 to 45 minutes.