1 1/3-inch slice peeled fresh ginger, coarsely chopped
2 tablespoons (packed) golden brown sugar
2 cups ice cubes
1/4 cup vodka
1/4 cup fresh lemon juice
2 tablespoons chile-pepper-flavored vodka (such as Absolut Peppar or Stolichnaya Pepper)
2 tablespoons Cointreau or other orange liqueur
Muddle ginger and sugar in cocktail shaker until paste forms. Add all remaining ingredients except crushed ice; cover and shake 15 seconds. Fill 2 old-fashioned glasses with crushed ice. Strain cocktail into glasses and serve.