Wash one medium cucumber under cold water and pat it dry completely. Cut it into chunks and run it through a vegetable juicer, or if you don't have one, grate it on the fine side of a box grater and press the shreds through cheesecloth or a fine mesh strainer over a bowl, squeezing firmly to extract as much liquid as possible.

Fill your margarita or coupe glass with ice water and set it in the freezer while you prepare the cocktail. A cold glass keeps your drink cold longer and makes each sip taste fresher instead of warming up as the ice melts.

Cut a lime in half and juice it using a citrus juicer or your hands, pressing firmly to extract maximum juice. Strain through a fine mesh strainer to catch any pulp or seeds that could cloud your drink.

Using a jigger or measuring spoon, carefully measure 2 ounces of silver tequila, 3 ounces of fresh cucumber juice, 1 ounce of fresh lime juice, and 0.5 ounce of agave nectar into your cocktail shaker. Precision here makes the difference between a balanced drink and one that's either too sweet or too strong.

Add a generous handful of ice to your shaker, enough that the ice fills about half the shaker's volume. The ice cools the liquid rapidly while also providing dilution as it melts, which softens the tequila's edge and helps all the flavors meld together.

Close your shaker tightly and shake vigorously for about 15 seconds, until the outside of the shaker feels frosty and cold to the touch. Vigorous shaking aerates the mixture and ensures proper chilling without over-diluting the drink.

Remove your glass from the freezer and discard the ice water. A properly chilled glass is already at the right temperature, so you don't need to fill it with ice again if you work quickly.

Place your fine mesh strainer over the chilled glass and pour the shaker contents through it, capturing any small ice chips or cucumber pulp that might have slipped through. Pour slowly and steadily to maintain the drink's silky texture, then garnish with a thin cucumber slice or a squeeze of fresh lime if you like.