Rinse the leeks in cold water very well to remove sand and dirt. Then dry well.
Heat the oil in a pot over medium heat. Once hot, sauté the leeks, garlic, onion, and celery until they start to soften. Crumble in the sausage and brown well while breaking it into smaller pieces. Stir in broth and potatoes. Season with salt and pepper.
Cover with a lid and simmer until potatoes are tender (about 15-20 minutes). Once the potatoes are tender, give a few pulses using an immersion blender or stand up blender to give the soup a creamy consistency. (Leave some of the potatoes chunky, be careful not to overblend)
Blending the potatoes will thicken the soup, stir in more broth to thin the soup out to a creamy consistency as needed. Season with salt and pepper to taste.
Drizzle with olive oil on top. Serve with crackers, garnish with leeks.