In a large dutch oven, melt butter over medium-high heat and then add onion and ½ teaspoon of salt. Cook, stirring frequently, until the onion is softened, about 3-4 minutes. Add the garlic and cook, stirring constantly, for 30 seconds. Add the water, ham steak, bacon, split peas, thyme, and bay leaves and increase heat to high. Bring soup to a simmer, stirring frequently. Once the soup is simmering, reduce heat to low and cover the pot. Cook for about 45 minutes, until the peas are tender but not falling apart.
Remove the ham steak to a plate and cover with foil.
Stir in the carrots and celery then cover the pot and continue to simmer about 30 minutes longer. The veggies will be tender and the peas will have broken down almost completely.
Shred the ham into small pieces, using two forks.
Remove the thyme, bay leaves, and bacon slices from the soup and discard. Add the ham to the soup and bring it back to a simmer. Taste the soup and add salt and pepper to your liking. Add some water if the soup is too thick, or allow to simmer uncovered if too thin. Serve, topped with fresh peas, fresh mint, and a drizzle of balsamic vinegar.
