Add the bacon to a large pot and cook on medium high heat, stirring occasionally until crisp. Use a slotted spoon to transfer the bacon to a plate lined with paper towels to remove extra grease. Set aside and do not clean the pot.
Add the carrot, celery, and shallots. Cook, stirring occasionally, until the shallots are soft and beginning to turn golden, about 10 minutes.
Add the chestnuts and chicken stock. Bring to a boil and then reduce heat to a simmer. Simmer for 30 minutes, or until the chestnuts are soft and tender.
Transfer the mixture to a blender and add the cream. Blend until completely smooth. Adjust salt to taste.
Serve with a spoonful of crème fraîche, reserved bacon, chives, and sprinkling of nutmeg, if desired.