Brushing red currant jelly onto the berries makes them shine, like those fruit tarts you see in bakeries. If you can't find red currant jelly, use strawberry jelly (don't subsitute jam or preserves).
Whisk together cornstarch, salt, and 3/4 cup of the sugar in a medium-size heavy saucepan. Whisk together half-and-half, heavy cream, egg yolks, and vanilla bean paste in a small bowl; gradually add to cornstarch mixture, whisking constantly.
Bring mixture just to a boil over medium, whisking constantly, about 15 minutes. Boil, whisking constantly, 1 minute. Remove from heat, and transfer to a bowl. Cover with plastic wrap, placing plastic wrap directly on mixture in bowl; chill 4 to 24 hours.
Preheat oven to 350°F. Pulse crushed cookies and chocolate in a food processor until finely ground, 8 to 10 times. Transfer to a bowl, and stir in butter and remaining 1/4 cup sugar. Firmly press mixture on bottom, up sides, and onto lip of a lightly greased (with cooking spray) 9-inch round pie pan. Bake in preheated oven 10 minutes. Transfer to a wire rack, and cool completely, about 30 minutes.
Spoon chilled half-and-half mixture into prepared crust. Freeze 2 hours. (Pie may be frozen up to 24 hours.)
Toss together strawberry halves, blueberries, blackberries, and raspberries in a medium bowl. Arrange berries on top of pie. Top with 3 whole strawberries. (If pie was frozen longer than 2 hours, top with berries, and let stand 1 hour before brushing with jelly mixture.)
Melt jelly in a small microwave-safe bowl at HIGH, stirring every 15 seconds, until melted, about 1 minute. Stir in water. Brush jelly mixture gently over berries.