Sweat the leeks. Coat the bottom of a large pot with olive oil and place over medium heat. When the oil is hot, add the chopped leeks. Cook, stirring occasionally, for about 5 minutes until they soften and become fragrant. Don't rush this step properly sweated leeks make a significant difference to the final flavor.
Add the garlic. Stir in the minced garlic and cook with the leeks for about 1 minute, stirring constantly, until very fragrant.
Add wine and thyme. Stir in the dry white wine and dried thyme. Bring to a simmer and let it cook for about 4 minutes until the wine reduces by roughly half. The alcohol cooks off completely, leaving behind a subtle acidity and depth that elevates the entire soup.
Add potatoes, broth, and coconut milk. Stir in the potato pieces, vegetable broth, and coconut milk. Raise the heat and bring everything to a simmer. Reduce the heat and cook uncovered for about 15 minutes until the potatoes are just fork-tender.
Remove from heat. Take the pot off the burner before blending.
Blend the soup. Use an immersion blender to blend directly in the pot, or transfer in batches to a countertop blender. Blend to your preferred consistency leave some chunks for a rustic texture, or blend completely smooth for an elegant, restaurant-style finish. Stop blending as soon as you reach the texture you want overblending potato soup makes it gummy and unpleasant.
Adjust consistency and seasoning. Return the soup to the pot if removed for blending. Add more vegetable broth if you prefer a thinner consistency. Reheat briefly if needed, then season generously with salt and pepper.
Serve. Ladle into bowls and finish with a scatter of fresh chives. Add crusty bread on the side and serve immediately.
