Preheat the oven to 350 degrees F. Cut the squash in half on the equator and remove the seeds with a spoon. Cut a flat spot on each end so the squash will sit flat. Line a baking sheet with aluminum foil and arrange the squash, cut side up. To 3 of the squash halves, add a peeled shallot and to the other 3 add 2 garlic cloves on each. Drizzle with 2 tablespoons of olive oil and season with 1 tablespoon each of the salt and freshly cracked pepper. Roast in the hot oven until very tender and starting to caramelize and collapse, approximately 1 hour. Remove from oven and when cool enough to handle, remove the squash from the skin. Reserve the roasted shallots and garlic with the squash. Can be done ahead.
In a large Dutch oven, heat 1 tablespoon of the olive oil and 2 tablespoons of the butter over medium-high heat and when the butter is starting to foam, add the raw diced shallots and saute until they are starting to caramelize, about 5 to 6 minutes. Deglaze with 1/2 cup of the chicken stock and stir to remove any fond. Reduce the heat to medium-low and add in the reserved squash, roasted shallots and garlic and then the remaining chicken stock. Stir to combine, then puree with a stick blender. The mixture will be very thick. Add in the cayenne, white pepper and the herbs. Stir in the cream and Worcestershire sauce and heat slowly over medium-low heat. When the mixture comes to a slow simmer, mix again with the stick blender and stir in 1/4 cup of the Parmesan and turn heat to low. Serve with a fresh crack of black pepper, a nice drizzle of extra-virgin olive oil and a light sprinkle of remaining Parmesan. Ladle into soup bowls and serve.