Bring the lime juice, sugar, cilantro stems, halved jalapeno and 1/4 cup water to a simmer in a small saucepan. Cook, stirring, until the sugar dissolves, about 1 minute. Transfer to a small bowl and refrigerate until chilled, at least 1 hour and up to 4 hours.
Meanwhile, puree the cucumbers in a blender. Strain the puree through a fine-mesh sieve into a liquid measuring cup, pressing on the solids with a rubber spatula. Refrigerate the cucumber puree until chilled.
Pulse the salt and cilantro leaves in a mini food processor until the salt is green; transfer to a small plate. Run the squeezed limes around the rims of 4 glasses, then dip in the cilantro salt to coat.
Strain the jalapeno syrup into a small pitcher. Stir in the cucumber juice, tequila and triple sec. Fill the prepared glasses with ice and add a few jalapeno slices. Pour in the cocktail.
Photograph by Linda Xiao