On a shallow plate, toss the sugar with the lime zest. Run a lime wedge around the rim of a cocktail glass, then dip and swirl into the lime sugar. Shake out excess.
In the bottom of a shaker, muddle the basil leaves, strawberry jam, and lime juice. Add the tequila, amaretto, and agave (if using). Fill with ice and sake until cold. Strain or simply pour into a cocktail glass and add extra ice as needed. Top with a splash of sparkling water.