Trim bottom end and woody top half of each lemongrass stalk, reserving tops to use as garnish. Roughly chop remaining segments and add to a mini food processor.
Roughly chop ginger (no need to peel) and add to food processor along with sugar.
Pulse until ingredients are very finely chopped.
Transfer lemongrass mixture to a 4-cup glass measuring cup, quart-sized mason jar, or nonreactive bowl.
Pour in the vodka and stir.
Let sit for at least 15 minutes at room temperature — up to a week in the fridge.
Strain vodka into a pitcher through a fine-mesh sieve, pressing down a bit to extract flavor from the solids.
Pour in sake, grapefruit juice, and Meyer lemon juice and give it a stir.
Add two cups of ice.
To serve, pour into glasses filled partway with ice. To garnish, trim woody bits from tops of reserved lemongrass stalks and slice to form a fanlike shape.