Heat olive oil over medium heat in a large soup pot. Add chopped leaks and sauté 5-6 minutes, stirring regularly until tender.
Add the minced garlic and continue sautéing, stirring regularly, for one minute.
Add the peeled, chopped potatoes, broth, salt, pepper and thyme. Bring to a boil, then reduce heat, cover, and simmer 20 minutes until potatoes are tender. Remove from heat.
Use an immersion blender to blend until smooth and creamy. Be careful not to over-mix as the potatoes can become gummy. (Note: this can also be done in batches in a regular high powered blender.)
Once the soup is smooth, add in the cream and blend to incorporate.