Place turnips in a pot and add water until covered by 3 to 4 inches; bring to a boil. Cook, stirring every 5 to 10 minutes, until soft, about 30 minutes. Drain.
Heat oil in the pot over medium heat. Add leeks and celery; cook and stir until slightly tender, about 5 minutes.
Blend leek mixture, 1 cup chicken stock, bacon, cornstarch, salt, and pepper in a food processor until smooth. Pour back into the pot.
Blend half of the turnips and 1 cup chicken stock in the food processor until smooth; pour into the pot. Repeat with remaining turnips and 1 cup chicken stock.
Pour remaining 1 cup chicken stock into the pot; stir in milk. Simmer, without boiling, until thickened, about 30 minutes. Ladle into bowls; garnish with green onions.
