Combine sugar, honey, salt, and 1/2 cup water in a medium saucepan. Bring to a boil. Simmer for 5 minutes, stirring frequently. Remove from heat and cool. Set aside.
Blend ginger and 2 tbsp water in a blender and puree until smooth. Pour through a strainer and set aside.
Combine flour, salt, and sesame oil in the bowl of a food processor. Pulse until the mixture resembles fine breadcrumbs. Add ginger juice, honey, rice wine, and 2 tbsp water and process until a smooth, pliable dough forms.
Turn the dough out onto a lightly floured surface and roll it out into a 1/4 inch thick circle, about 12 inches wide. Using a small cookie cutter, cut dough into 20 cookies.
Heat 2 inches of oil in a 6 qt saucepan until a deep-fry thermometer reads 300 degrees F. Working in batches, carefully slide cookies into oil and fry, flipping once, until puffed and golden (3-4 minutes).
Using a slotted spoon, transfer to a baking sheet with a wire rack. Dip cookies into syrup and transfer to a serving plate. Dust with cinnamon and pine nuts, if using.