To a small pot over medium heat, add the butter and cook, using a wooden spoon to stir frequently, until the foam subsides and the butter begins to brown, with specks of darker brown collecting at the bottom of the pan, 5–6 minutes. Remove from heat and continue stirring to cool slightly and stop the cooking. Set aside to cool to room temperature.
Preheat an oven (with one of its racks positioned in the center) to 300°F. To a food processor, add the walnuts and pulse to chop finely. Transfer the nuts to a medium bowl. Add the biscuits to the food processor, pulse them until finely chopped, and transfer them to the bowl as well. Add the sugar and salt and stir well to combine then add the browned butter, mixing until the dry ingredients are thoroughly moistened. Lightly butter a 9-inch glass or aluminum pie plate. Transfer the crumb mixture to the pie plate, then use your fingertips to press it into an even layer, working the mixture into the corners of the plate and evenly up the sides. Cover the crust with a piece of plastic wrap and use the bottom of a measuring cup or glass to smooth and compress the crumbs into place. Transfer to the freezer until the dough is firm and set, at least 15 minutes or overnight.
Retrieve the pie shell from the freezer, place it on a large baking sheet, remove the plastic film, then immediately transfer to the oven and bake until just set and lightly browned, 12–14 minutes. Set aside to cool completely.
Meanwhile, soften the ice cream: Transfer the container from the freezer to the fridge and allow it to soften slightly, about 20 minutes. When the crust is cooled, use a large silicone spatula to fill the pie plate, gently smoothing the ice cream in place without breaking apart the crust. Cover the pie with plastic wrap and return it to the freezer until the ice cream is firm but still scoopable, 15–20 minutes.
Retrieve the pie from the freezer and remove the plastic wrap. Use a long knife to score the surface into 8 equal portions. Starting 1” from the edge of the pie, use a melon baller or small scoop to hollow out eight 1-inch holes, (without going to the very bottom of the pie) one per portion. Re-cover the pie with plastic wrap and return the pie to the freezer until the ice cream is completely firm about 1 hour.
To a large chilled bowl of a stand mixer fitted with the whip attachment, add the heavy cream and confectioners’ sugar and beat until thickened. Add the dissolved instant espresso powder and vanilla extract, beating just until the mixture thickens and holds soft peaks. Finish whisking the cream to stiff peaks by hand, being careful not to overwhip. Cover the cream and refrigerate until ready to serve.
To serve the pie. Measure the rum or Cognac into a liquid measuring cup with a spout. Retrieve the pie from the freezer and remove the plastic wrap. If necessary, use the melon baller to make distinctive hollows- sometimes the ice cream softens and needs a little definition. Carefully fill each hollow with 1 tablespoon of the liquor, taking care not to overfill. Place a walnut half at the crust edge of each portion so you will know where to slice the pie. Retrieve the whipping cream from the refrigerator and cover the top of the pie with the cream. Garnish the pie with chocolate shavings and additional toasted walnuts. Use a hot knife to slice carefully, transfer to rimmed dessert plates or small, shallow bowls, and serve immediately. Cover and freeze any leftovers for up to 3 days.