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Line a 10-inch bundt pan with plastic wrap, leaving some overhanging the edge.
Trim a 3/4-inch-thick slice from the bottom of the chocolate cake. Transfer to a cake board and cut a hole in the center using a 1 1/2-inch round cutter. Set aside.
Cut half of the remaining cake into wedges. Add it to a food processor or a stand mixer with the softened vanilla ice cream. Process until the two are combined. Pour the ice cream mixture into the prepared Bundt pan and spread in an even layer. Freeze until mostly solid, about 2 hours. Any remaining cake can be kept in the freezer and blended with ice cream for a quick dessert.
Once the ice cream has set, gently release it from the pan by pulling up the plastic wrap. Flip the ice cream over onto the 3/4-inch-thick slice of cake, lining up the holes in the ice cream and cake. Remove the plastic wrap and place the cake back in the freezer while you make the meringue.
For the Italian meringue: In the bowl of a stand mixer fitted with a whip attachment, whip the egg whites on medium until soft peaks form, 3 to 4 minutes.
Meanwhile, in a small, high-sided pot with a candy thermometer attached, combine the sugar and 1/3 cup of water. Place over medium-high heat and cook, without stirring, until the sugar syrup reads soft ball stage on the candy thermometer, 235 to 240 degrees F, 5 to 10 minutes.
As soon as the syrup reaches soft ball stage, turn the mixer on low and carefully stream the syrup into the whipped egg whites. Beat until the whites are stiff and glossy and the bowl is no longer hot, about 2 minutes.
Remove the ice cream cake from the freezer and cover it with the meringue using a rubber or offset spatula. Finish by using a kitchen blowtorch to brown the meringue.