Melt butter and oil in a stockpot or Dutch oven over medium-low heat. Add celery, carrot, onion, and salt; cook and stir until tender, about 5 minutes. Stir in peas and garlic.

Nestle ham bone into the pot; add stock and diced ham. Add enough water so that liquid is about 1 1/2 inches from the top of the pot; bring to a boil.

Reduce heat to low and cover. Simmer, stirring every 30 minutes and scraping the bottom of the pot, until peas are mostly disintegrated, about 2 1/2 hours. Turn ham bone and skim the surface for fat once every hour.

Remove ham bone from the pot and place on a cutting board until cool enough to handle. Remove any meat from the bones and cut into bite-sized pieces; stir into soup until warmed through and season with salt and pepper.