Rinse the Dried Yellow Split Peas under cold water and pick out any debris; drain well.
Place the rinsed peas in a large saucepan and add the Ham Stock (use roughly 3–4 cups of stock per cup of peas as a guide). Bring to a boil, then reduce heat to maintain a gentle simmer.
Add a chopped Onion and a Bay Leaf to the pot. Skim any foam from the surface, cover partially, and simmer gently until the peas are completely soft and beginning to break down, about 45–60 minutes (stir occasionally and add extra stock or water if the mixture looks too dry).
Remove and discard the Bay Leaf. Mash the peas in the pot with a potato masher or blend briefly with an immersion blender to your preferred texture—smooth or slightly rustic.
Stir in the Unsalted Butter until melted and incorporated, then season to taste with Salt and freshly ground Black Pepper. If the pudding is too thick, loosen with a splash of hot stock or water and heat through for a few minutes to meld the flavors.
Serve the pease pudding warm or at room temperature as a spread for bread, a side to roasted or cured meats, or simply with pickles and mustard for a classic presentation.
