Whisk white sugar, cornstarch, and salt together in a heavy saucepan over medium heat. Gradually add eggnog to sugar mixture, stirring constantly. Cook and stir until eggnog mixture is thickened, 3 to 4 minutes; continue to cook for 2 minutes more. Remove from heat.
Stir gelatin mixture into eggnog mixture until gelatin is completely dissolved. Stir in rum extract. Chill eggnog mixture in the refrigerator until cold, at least 1 hour.
Beat 1 cup cream in a cold glass or metal bowl until stiff peaks form. Lift your beater or whisk straight up: the cream will form sharp peaks. Fold whipped cream into chilled eggnog mixture; pour mixture into prepared cookie crust. Chill pie in the refrigerator until set, at least 1 hour.
Beat cream in a cold glass or metal bowl. Gradually add confectioners' sugar, continuing to beat until soft peaks form. Lift your beater or whisk straight up: the cream will form soft mounds rather than a sharp peak. Spread cream atop the chilled pie. Refrigerate until chilled, at least 1 hour.