For my birthday last year, a friend gave me an entire decadent pound of caviar and it inspired me to create a meal around it: caviar with blinis and creme fraiche; a whole smoked salmon; and dozens of fresh oysters topped with this zesty, refreshing granita. I like how its icy texture plays against the oysters' creamy one and how it it enhances--rather than overpowers--their delicate, saline flavor. My favorites are freshly shucked Kumamotos. —gluttonforlife
Peel cucumbers, cut in half and remove seeds. Coarsely chop into chunks and process in blender with half the jalapeño and the rest of the ingredients. Press liquid through fine strainer. Taste and adjust seasonings.
Unwrap and stir with a fork, scraping to break up the icy bits around the edges. Repeat this every 20 to 30 minutes until there is no liquid left. This will take approximately 2-3 hours.
To serve, scrape granita into a bowl. Spoon a teaspoon or so onto each oyster and top with a bit more minced jalapeño, if desired.