Crushing ice, refreezing it, and crushing it again gives it a texture like that of a snow cone.
Puree lychees with 1/4 cup syrup in a food processor or blender until smooth. Transfer to a sieve set over a bowl; stir and press on solids to drain as much liquid as possible. (You should have about 1 cup.) Discard solids and refrigerate juice until cold, at least 1 hour; juice can be made up to 3 days ahead.
Bring remaining syrup and ginger to a boil in a small saucepan over high heat. Reduce heat and simmer 3 minutes. Let cool to room temperature, strain, and discard ginger. Refrigerate syrup at least 1 hour; syrup can be made up to 3 days ahead.
Pulse ice, 2 cups at a time, in food processor. Transfer to a freezer bag and freeze at least 30 minutes and up to 1 day.
Stir together lychee juice and ginger syrup. Break ice in bag into small pieces with the bottom of a heavy skillet. Pulse ice, 2 cups at a time, in processor to the texture of shaved ice. Divide ice and lychee mixture evenly among bowls and top with liqueur. Serve with cookies.