Ginger beer is a grown-up, non-alcoholic version of ginger ale that's very popular in Britain. Now it's starting to appear on grocery shelves in US specialty stores, but it's fun, easy, and less expensive to make at home. You don't need to peel the ginger, and the quickest way to chop it is with a food processor. Vary the "heat" and intensity of this beverage by adding more or less sugar syrup and seltzer water. Extra ginger juice can be refrigerated for up to 1 week or frozen for 3 months.
Combine the chopped ginger with 3 cups of water in a small saucepan. Bring to a boil over high heat and cook for 1 minute. Reduce the heat to low and continue to cook for 10 minutes. Remove pan from the heat and let the mixture steep at room temperature for at least 3 hours, or overnight.
In another small saucepan, combine the sugar with 1 cup of water. Bring to a boil over high heat, stirring to dissolve the sugar. Cook the syrup for 2 minutes, then remove the pan from the heat and let cool. Pour the syrup into a clean container, cover, and refrigerate until cold.
After steeping, strain the ginger juice through a fine mesh sieve, pressing hard on the solids to extract all of the liquid. Discard the solids and refrigerate the juice until cold.
Combine 1 cup of ginger juice with 1 cup of sugar syrup in a pitcher and stir to blend. Add the juice of 2 limes and stir again. To serve, fill 4 tall glasses with crushed ice. Divide the ginger beer among the glasses and top with seltzer water. Stir each drink to blend and serve immediately.