Coat the bottom of a large pot with the oil and place it over medium heat. Give the oil a minute to heat up, then add the onion, carrots, parsnips, and fennel. Cook the vegetables for about 10 minutes, stirring occasionally, until they begin to soften. Stir in the garlic and cook it with the vegetables for about 1 minute, stirring constantly, until it becomes very fragrant.
Add the wine to the pot and bring it to a simmer. Let it simmer until the volume has reduced by about half, about 4 minutes, stirring occasionally.
Stir in the broth, sweet potato, rutabaga, rosemary, and thyme. Raise the heat to high and bring the broth to a boil, then reduce the heat and let the soup simmer until the vegetables are fork tender, about 20 minutes.
Remove the pot from heat. Remove and discard the rosemary sprig. Blend the soup until smooth and creamy. You can leave it in the pot and use a hand blender, or transfer it in batches to a conventional blender, then return to the pot. Always use caution when blending hot liquids.
Stir in the brown sugar, nutmeg, and coconut milk. Warm the soup back up over the burner if needed, then turn off the heat, give it a taste-test, and season the soup with salt and pepper to taste.
Ladle into bowls, serve, and enjoy.