Make the shrub: In a medium bowl, combine the blueberries, sugar and jalapeño slices. Toss to coat with sugar and mash to begin to release the juices from the blueberries. Transfer to a sealable container and refrigerate for at least 2 hours and up to 48 hours, shaking the container halfway through.
In another sealable container, combine the remaining shrub ingredients and store in the fridge while the berries macerate.
After 2 days (or less if you're pressed for time), strain both the macerated berries and the vinegar, discarding the solids. Combine the strained vinegar and syrup to make the shrub; makes about 3 cups.
Make the cocktail: In an ice-filled glass, combine the shrub, gin and lime juice. Stir until chilled, then strain into a chilled coupe. Top with soda, garnish with a jalapeño slice, then serve.