In a large pot or Dutch oven, heat the olive oil over medium heat. Add the chopped onion, carrots, and celery. Sauté for about 5 minutes, or until the vegetables begin to soften.
Stir in the minced garlic and cook for another minute until fragrant.
Add the rinsed split peas, ham hock (or diced ham), chicken broth, and dried thyme to the pot. Bring the mixture to a boil.
Once boiling, reduce the heat to low, cover, and simmer for about 45 minutes, or until the peas are tender. If using a ham hock, remove it after cooking and shred any meat back into the soup.
Season the soup with salt and pepper to taste. Stir well and keep warm while you make the cornbread.
Preheat the oven to 400°F (200°C).
In a mixing bowl, combine the cornmeal, flour, baking powder, salt, and sugar.
In another bowl, whisk together the milk, eggs, and melted butter. Pour the wet ingredients into the dry ingredients, stirring until just combined.
Pour the cornbread batter into an oven-safe baking dish or a cast-iron skillet. Bake for 20-25 minutes or until the top is golden and a toothpick inserted in the center comes out clean.
Once the cornbread is finished, remove it from the oven and let it cool slightly.
Serve the warm Split Pea & Ham Soup in bowls with slices of cornbread on the side.