Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper. Whisk the flour, baking soda and salt in a medium bowl. Beat the butter, granulated sugar and brown sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 minutes. Beat in the egg, lemon zest, lemon juice and vanilla. Reduce the speed to low; beat in the flour mixture in two batches, alternating with the sour cream.
Roll the dough into 16 balls. Arrange on the baking sheets; sprinkle with sanding sugar. Bake until the edges are set, 12 to 15 minutes. Let cool 5 minutes on the pans, then transfer to racks to cool completely.
Sandwich the ice cream between the cookies. Freeze until firm, 3 hours.