Bring cream, milk, 1/2 cup sugar, and salt to a boil in a large heavy saucepan over medium heat, stirring to dissolve sugar; remove from heat. Whisk eggs yolks and remaining 1/2 cup sugar in a large bowl to blend; gradually whisk in hot milk mixture and return to saucepan. Stir over low heat until custard thickens and your finger leaves a path on the back of a spoon when drawn across, about 6 minutes. Pour through a fine mesh strainer into a medium bowl of ice water; stir until cold.
Process custard in an ice cream maker according to manufacturer’s instructions. Transfer to a container, cover, and freeze.
Do ahead: Can be made 3 days ahead.
Preheat oven to 250 degrees F. Line a baking sheet with parchment paper. Whisk sugar, egg white, and salt in a medium bowl just until bubbles form. Stir in hazelnuts. Spread out on prepared sheet. Bake, stirring every 10 minutes, until nuts are golden brown and caramelized, about 50 minutes.
Do ahead: Can be made 1 day ahead. Store in airtight at room temperature.
Preheat oven to 325 degrees F. Coat tart pan with nonstick spray. Finely grind cookies in a food processor. Drizzle butter over; pulse until blended. Press evenly onto bottom and up sides of pan. Bake until firm, about 15 minutes. Let cool completely.
Place chocolate in a medium bowl. Bring cream to a boil in a small saucepan. Pour over chocolate; stir until melted. Pour filling over cooled crust. Freeze until firm, at least 2 hours and up to 2 days.
Remove the tart pan side. Cut tart into wedges, working quickly as tart will become gooey as it sits; place 1 piece on each plate. Serve with a scoop of black pepper ice cream and garnish with candied hazelnuts.