In a small bowl, whisk egg with vanilla. Set aside.
In a food processor, pulse hazelnuts and sugar until finely ground. Add flour and salt; pulse again.
Add butter and pulse until coarse crumbs form. While running, add egg mixture and process until dough forms.
Turn dough onto plastic wrap, shape into a disk, and refrigerate for 1 hour.
Roll dough to an 11-inch circle, fit into 9-inch tart pan, trim edges, and patch any weak spots. Freeze shell for 30 minutes.
Preheat oven to 375°F (190°C). Line shell with foil and weights. Bake 30 minutes, remove foil, bake 5–10 more minutes until golden. Cool on wire rack.
Toast chopped hazelnuts and sugar in a skillet until golden and caramelized. Set aside.
Heat cream to a simmer, remove from heat. Stir in chocolate and butter, cover 2 minutes. Stir until smooth.
Add Nutella and stir until glossy. Pour into cooled crust over candied nuts. Smooth top.
Cover and chill 2 hours or until set.
For glaze: whisk Nutella and milk until smooth, warm slightly if needed. Spread over set tart.
Remove tart ring, slice and serve chilled or at room temperature.