Preheat oven to 375 degrees . Lightly grease a 10x15 inch jellyroll pan. Line with parchment, then grease the parchment and the sides of the pan. Using an electric mixer, beat the butter, sugar and salt on medium speed until fluffy, about 3 minutes. Beat in the vanilla, then the eggs 1 at a time. Alternately add the flour and milk in 3 additions. Fold in the nuts. Pour the batter into the prepared pan and bake until light golden and a toothpick inserted into the center comes out clean, 15-17 minutes. Allow the cake to cool slightly in the pan, then invert onto a parchment-lined rack to cool completely
Transfer the cake to a work surface and cut in half crosswise. With a sheet of parchment between the layers, stack the cake halves, wrap in plastic and freeze until firm, about 30 minutes.
Let the ice cream soften in the refrigerator until spreadable. Remove 1 of the cake layers from the freezer and cover with an even layer of the softened vanilla ice cream, spreading all the way to the edges of the cake. Transfer to the freezer for 15 minutes. Repeat with the strawberry ice cream, then the pistachio ice cream. Top with the second cake layer and freeze until set, at least 1 hour.
Remove the cake from the freezer and trim the edges. Cut in half crosswise, then cut each half into 6 equal rectangles, making 12 bars. Serve, or wrap and freeze up to 2 weeks.