1. Make the ginger cordial: In a medium saucepan set over high heat, bring the water, brown sugar, ginger coins and lime zest to a boil. Turn off the heat, cover and steep for 30 minutes. Strain through a fine-mesh sieve and into a medium bowl. Add the ¼ cup of the lime juice, cover the bowl with plastic wrap and refrigerate until chilled, at least 2 hours.

2. Make the spicy ginger: If using a juicer, juice the ginger. If using a Microplane grater, grate the ginger, then place the pulp in a fine-mesh sieve set over a medium bowl. Press on the sieve to extract the juices. In a small saucepan set over medium heat, combine the ginger juice and turbinado sugar, stirring often, until the sugar is dissolved, about 1 minute. Turn off the heat and pour into a small bowl. Refrigerate until chilled.

3. Make the mixer: In a cocktail pitcher, add the spicy ginger, 5 ounces ginger-lime cordial and the remaining lime juice. Stir to combine. Serve on a tray with glasses, a bottle of club soda, a bottle of gin and a bucket of ice.