Preparation

Make the ice cream

In a high-powered blender, combine the blackberries, sugar, buttermilk, lemon juice, vodka, and salt. Blend until smooth. Strain through a fine-mesh sieve, pushing with the back of a rubber spatula, to remove the seeds. Return to the cleaned blender, add the heavy cream, and blend until thick, 1 minute. Churn in an ice cream maker according to manufacturer’s instructions, and transfer to a freezer container to freeze overnight.

Make the cookies

In a small saucepan, melt the butter over high heat. Reduce the heat to medium and continue to cook, swirling constantly, until deep amber brown and foamy, 3 to 4 minutes. Remove from the heat, transfer to a bowl, and refrigerate until chilled and thickened slightly, but still pourable, about 30 minutes.

Preheat the oven to 350°F. Transfer the chilled brown butter to a large bowl, and add the brown sugar, granulated sugar, vanilla, and salt. Using an electric mixer, beat on medium speed until combined. Add the egg and beat until smooth. In a small bowl, whisk together the flour and baking soda. Add to the butter mixture and beat until combined. Using an ice cream scoop (about 3 Tbs.), drop mounds of dough on a cookie sheet, spacing them at least 3 inches apart. Bake for 8 minutes, rotate the pan, and bake until lightly golden brown around the edges and just set in the center, 5 to 7 minutes. Remove, let cool on a wire rack, then transfer to the freezer to harden.

Assemble the sandwiches

Remove the ice cream and cookies from the freezer. Turn over 5 of the cookies, and using a large ice cream scoop, divide the ice cream on top (about 3 to 4 Tbs. per sandwich). Top with the remaining 5 cookies, pressing down and smoothing the edges. Refreeze at least 2 hours.

Glaze the sandwiches

Place the lychees in a food processor, and process until very smooth. Strain through a fine-mesh sieve into a small saucepan, pressing with the back of a rubber spatula to get all the juices (about 1/2 cup). Add the powdered sugar to the pan, and cook over high heat until thickened, stirring constantly, 3 to 5 minutes. Remove from the heat, and whisk in the butter. Transfer to a small bowl, and let cool completely, about 1 hour.

Remove the sandwiches from the freezer, and partially dip them in the cooled glaze. Sprinkle the glaze liberally with sesame seeds, and return the sandwiches to the cookie sheet and freeze at least 20 minutes.