Heat the olive oil in a large pot over medium heat. Add the sliced leeks and minced garlic, then cook for 3 to 4 minutes until the leeks soften.
Add the cauliflower florets and vegetable broth to the pot. Bring the mixture to a boil, then reduce the heat and simmer for 15 minutes until the cauliflower is tender.
Remove the pot from the heat. Use an immersion blender to puree the soup until smooth, or transfer it in batches to a countertop blender.
Return the pot to low heat. Stir in the milk and season with salt and black pepper. Heat for 2 minutes, stirring occasionally, until warmed through.
Ladle the soup into bowls and top each serving with chopped chives.