Sauté the Leeks: Warm the olive oil or vegan butter in a large, heavy-bottomed pot over medium heat. Add the sliced leeks and cook for about 5 minutes, stirring occasionally, until they soften and turn lightly golden.
Build the Flavor Base: Stir in the minced garlic, cauliflower florets, diced potatoes, and thyme. Allow everything to cook together for another 3 minutes, releasing the aromas and slightly toasting the garlic.
Simmer the Vegetables: Pour in the vegetable broth, scraping the bottom of the pot to loosen any caramelized bits. Bring the mixture to a gentle boil, then reduce the heat to a simmer. Cook uncovered for 18–20 minutes, or until the cauliflower and potatoes are tender when pierced with a fork.
Blend to Creamy Perfection: Remove the pot from the heat and let it cool slightly for safety. Using either a countertop blender (working in batches) or an immersion blender directly in the pot, puree the soup until it reaches a smooth, velvety consistency.
Adjust and Serve: Return the blended soup to the pot if needed, reheat gently over medium-low heat, and season with salt and pepper to your liking. Ladle into bowls and garnish with fresh chives, shredded plant-based cheese, or croutons before serving.