Preheat the oven to 350°F (175°C).

In a medium mixing bowl, combine the cookie crumbs and the 1/4 cup of granulated sugar. Pour the melted butter over the crumb mixture and stir until everything is evenly moistened and resembles wet sand.

Press the mixture firmly and evenly into the bottom and up the sides of a 9-inch deep-dish pie plate. Use the bottom of a measuring cup to help compact the crust.

Bake the crust for 8 to 10 minutes until set. Remove from the oven and allow the crust to cool completely on a wire rack before proceeding.

In the bowl of a stand mixer fitted with the paddle attachment, cream the softened 1 cup of butter on medium speed until smooth, about 1 minute.

Gradually add the 1 1/2 cups of sifted powdered sugar to the butter and beat on medium-high speed until the mixture is light, fluffy, and very pale in color, scraping down the sides of the bowl often. This process should take approximately 4 to 5 minutes.

Add the melted and cooled chocolate and the vanilla extract to the creamed mixture. Beat on low speed until just combined, then increase to medium speed and beat for 1 minute.

Ensure the mixture is fully incorporated, then stop the mixer and scrape down the sides. Add the eggs one at a time, beating well after each addition.

Once all four eggs have been added, increase the mixer speed to high and beat the filling vigorously for a full 5 minutes. This long beating time is crucial for achieving the light, airy, and 'silky' texture. The mixture will lighten in color and volume significantly.

Carefully spoon the prepared chocolate silk filling into the cooled pie crust. Use a rubber spatula or the back of a spoon to smooth the top of the filling.

Cover the pie loosely with plastic wrap or foil (avoid touching the filling surface). Refrigerate the pie for a minimum of 6 hours, or preferably overnight, until the filling is firm and fully set.

Just before serving, place the cold heavy whipping cream, 1/4 cup powdered sugar, and 1/2 teaspoon vanilla extract in the bowl of a stand mixer fitted with the whisk attachment.

Beat on medium-high speed until stiff peaks form. Be careful not to over-beat, which can turn the cream into butter.

Spread or pipe the freshly whipped cream generously over the chilled chocolate silk pie. Garnish with chocolate shavings, if desired.

Slice and serve immediately. Store leftovers tightly covered in the refrigerator.