Heat the olive oil in a large pot or Dutch oven over medium heat. Add the onion, carrots, and celery and cook, stirring occasionally, for 6 to 8 minutes, until the onion is soft. Stir in the garlic and cook 1 minute more.
Add the split peas, broth, ham, smoked paprika, thyme, 1 tsp salt, and 1/2 tsp pepper. Raise the heat and bring to a boil.
Cover, reduce the heat to low, and simmer for 60 to 75 minutes, stirring every 15 minutes, until the peas are fully tender and the soup is thick. If it gets too thick before the peas are tender, stir in a splash of broth or water.
Remove from the heat and taste for salt and pepper. For a creamier texture, mash some of the peas against the side of the pot or give the soup a few quick pulses with an immersion blender.
Ladle into bowls and serve hot. Cool leftovers and refrigerate up to 4 days, or freeze up to 3 months; loosen with broth when reheating.