Apple-Ginger-Pear Martinis

1 ½ cups pear nectar
2 ½ cups apple juice
4 limes
4 teaspoons honey
1 ¼ cups vodka
4-inch piece fresh ginger, peeled

Stir pear nectar and apple juice together in a pitcher. Halve 3 limes and squeeze 6 tablespoons juice into the pitcher. Add the honey and stir until dissolved; stir in the vodka. Grate ginger on the large holes of a box grater. Place the grated ginger in a paper towel and squeeze to extract 1 tablespoon of juice; add to pitcher. Stir, cover, and chill well. 

Pour into chilled martini glasses or serve over ice. Garnish each with a thin slice of the remaining lime.