In a pot over medium heat, melt the butter. Add the onions and garlic, cook until tender, about 3 minutes. Add the broth, the bring to a simmer.
Add the artichokes hearts and cook for 10 minutes. Add the potatoes, season with salt and pepper. Cover and cook for 30 minutes more.
Using an immersion blender, puree the soup until smooth. Strain the soup into a saucepan and add more broth if necessary, then add the cream. Serve the soup in bowls drizzled with extra cream and potato sticks.