To Prepare the Pumpkin:
In a large skillet, heat the olive oil over medium-high heat. Add the pumpkin and onion and stir to coat with oil. Reduce the heat to medium-low and cook until the pumpkin begins to soften and the onions are translucent but not brown, about 5 minutes. Pour in 1 cup of the chicken stock, or enough to cover the pumpkin slices, and continue cooking until the pumpkin is very soft, about 10 minutes. Remove from the heat and let cool slightly, then pass through a food mill (or press through a strainer with the back of a spoon) into a medium bowl. Reserve.

To Prepare the Garnish:
Heat the olive oil in a medium skillet over medium heat. Stir in the girolles and shallot and cook, stirring frequently, until mushrooms give off their liquid and it evaporates. Stir in 2 tablespoons of the butter to coat the mushrroms, then add the chives. Stir to combine and remove from the heat.

Put the bacon cubes in a small saucepan and cover with cold water. Bring to a boil over high heat, reduce the heat to medium-low, and simmer for 1 minute to blanch and remove some of the salt. Drain and rinse briefly in cold water. Drain again and pat the lardons dry.

Saute the lardons in a small skillet until they are brown and crisp. Transfer to paper towels to drain.

To Finish the Soup:
In a large saucepan, combine the pumpkin mixture with the remaining 1 cup chicken stock and stir to combine. Bring the mixture to a gentle boil, then stir in the 6 tablespoons of butter, salt, pepper, and lightly whipped cream. Pour the mixture into the bowl of an electric mixer and beat on medium speed until smooth and creamy. Adjust the seasoning to taste.

Pour into a warmed soup tureen. Divide the girolle mushroom mixture among four warmed soup plates. Spoon in the pumpkin soup. Sprinkle on the lardons, then, if you wish, place a dollop of whipped cream in the center of each serving. Garnish with the sprigs of chervil. Serve immediately.