Place a large pot over medium heat and pour in the olive oil. Once hot, add the butternut squash, onion, celery, carrots, and apple. Season with salt and pepper.
In a large mixing bowl, whisk together chicken stock and pumpkin until completely combined. Pour this mixture into the pot.
Add the brown sugar and water to the pot and stir everything together. Bring to a boil, then reduce heat to medium-low so the soup simmers.
Cook for 2 hours, uncovered, or until the liquid has reduced by 4 cups.
Using an immersion blender, carefully puree the soup until it's smooth. Stir in the milk until combined.
Portion the soup into bowls. If desired, top with a swirl of sour cream and some thyme leaves for garnish. Serve warm.