Melt butter in 3-quart pot over medium heat.
Add garlic, onion, and leeks. Cook until softened (about 5 minutes).
Add potatoes, salt, pepper, and cayenne. Stir ingredients together, cover pot and cook an additional 5 minutes, or until potatoes begin to soften (about 5 minutes).
Add broth, stir and bring to a boil. Reduce heat to a slow simmer. Cook until potatoes are completely soft (about 10 minutes).
Either use an immersion blender, or ladle soup into a big food processor or blender. Blend until pureed and return to pot. You may have to work in batches, depending on how big your appliance is.
Once pureed soup is returned to pot, reduce heat to low and stir in half and half. Cook for an additional 3-5 minutes or until hot.
Top with chives and serve.