Heat olive oil over medium heat in a large stockpot. Add onion; cook and stir until softened and translucent, 3 to 4 minutes.

Place potatoes, ham, carrots, celery, parsley, and Italian seasoning into pot. Cook and stir until vegetables begin to soften, about 5 minutes. Add bay leaves, chicken broth, corn, and peas and bring to a boil, about 5 minutes. Reduce heat to medium and cook until vegetables have cooked down and are tender, about 20 minutes.

Remove bay leaves; add butter and pepper to taste. Top with shredded Cheddar cheese.