In a colander, rinse split peas under running water and pick through them to discard any small stones or other debris. Set aside.
Heat the olive oil in a large soup pot or heavy-bottomed Dutch oven over medium-high heat.
Add the onion, carrots, and celery. Cook, stirring occasionally, until the onions turn translucent.
Stir in the garlic, salt, black pepper, and allspice. Cook for 2 minutes just until fragrant.
Add the split peas, stirring to coat the ingredients.
Pour in 6 cups of the chicken stock, add the thyme and bay leaves, and nestle the ham bone into the pot.
Increase the heat to high, bring the soup to a boil, then reduce to low, cover, and simmer for 45 to 50 minutes, until the split peas are very soft and beginning to break down.
Meanwhile, combine the remaining 2 cups of stock with the frozen green peas in a large bowl. Blend using an immersion (stick) blender until completely smooth and bright green.
Discard the thyme sprigs and bay leaves, then remove the ham bone. Stir in the green pea purée.
Cook for a few minutes more, stirring until heated through. Pull any meat off the bone, chop it, and add it back to the soup. Heat until the ham is warm. Season to taste, and then serve warm.