Peel the cucumber if the skin is thick or waxed; otherwise leave it on. Trim the ends and roughly chop the cucumber into 1–2 cm chunks. Seed the jalapeño (slice lengthwise, scrape out seeds) if you want milder heat, or leave some seeds for more spice; roughly chop.

Blend

Place the chopped cucumber, chopped jalapeño, lime juice, granulated sugar, cold water, and kosher salt into a blender. Blend on high until completely smooth, about 30–45 seconds.

Strain

Pour the blended mixture through a fine-mesh sieve or cheesecloth into a bowl, pressing or stirring with the back of a spoon to extract as much liquid as possible. Discard solids. Taste the strained juice and adjust: add up to 1 tbs more sugar if too tart, or a pinch more salt if flat.

Chill (optional)

If not using immediately, cover and refrigerate the cucumber-jalapeño base for 20–30 minutes so flavors mellow and the liquid is cold.

To Assemble

Prepare glasses

Fill 4 tall glasses with ice cubes. Add 2 cucumber slices and 1 jalapeño slice to each glass and tuck in a couple of mint leaves if using.

Mix spritzers

Measure 90 ml (about 3 fl oz) of the strained cucumber-jalapeño base into each glass. Top each with 150 ml (about 5 fl oz) club soda or sparkling water. Gently stir once to combine, keeping bubbles lively.

Adjust and garnish

Taste one spritzer and adjust sweetness or acidity by adding a little extra lime juice (tsp increments) or a pinch of sugar dissolved in a teaspoon of warm water. Garnish each glass with an additional cucumber slice and a jalapeño slice on the rim, and bruise a mint leaf on top if desired.

Serve

Serve immediately while fizzy. For a mocktail-to-cocktail option, add 30–45 ml tequila or gin per glass if desired.