Preheat the oven to 400°F/200°C. Line a baking sheet with parchment or foil. Arrange pumpkin, squash, carrots, garlic, and onion on the sheet. Drizzle with olive oil, season with salt and pepper, and roast until tender and caramelized, 35 to 45 minutes.
Meanwhile, cook apples with brown sugar and ginger in a small saucepan over medium heat until softened but not mushy, 5 to 7 minutes.
Squeeze roasted garlic from skins. Scoop pumpkin and squash flesh from their skins and transfer to a blender or large pot with roasted carrots, onion, and cooked apples. Add stock and blend until smooth. Strain if desired for a silky texture.
Pour soup into a Dutch oven. Stir in curry powder, thyme, bay leaves, cinnamon, nutmeg, salt, and pepper. Simmer over medium-low heat for 10 minutes. Stir in heavy cream and adjust seasoning.
Ladle into bowls. Drizzle with extra cream and sprinkle with chives and red pepper flakes. Serve with toasted baguette or grilled cheese.