In a large bowl, whisk the eggs, then whisk in the milk, vanilla and 1/4 tsp. salt. Toss with the babka and 1 cup chocolate chips, pressing to submerge babka. Cover and refrigerate overnight.
Preheat the oven to 350 degrees. Transfer the babka mixture to a greased 2 1/2-quart baking dish. Sprinkle with the remaining 1/2 cup chocolate chips and the hazelnuts. Using a pastry brush, dab the cream all over the top; sprinkle with the sugar. Bake until the custard is set in the center and the top is browned in spots, about 1 hour. Let cool, about 15 minutes.
In a medium bowl, whisk the melted ice cream, brandy and cinnamon. Serve with the warm bread pudding.