Special equipment: a cocktail shaker
Heat sugar and water in a 1-qt. saucepan over high heat, stirring, until sugar is dissolved. Then pour into a heatproof bowl set inside a large bowl of ice and cold water.
Let stand, stirring occasionally, until syrup is cold, about 3 minutes.
Purée the lychees with sugar syrup and lemon juice in a blender until smooth. Now force through a fine-mesh sieve into a bowl, pressing hard on solids and then discarding them.
Fill cocktail shaker halfway with ice cubes and add lychee purée, vodka, and Cointreau. Shake 15 seconds and strain into Martini glasses.