Preheat oven to 375 degrees F (190 degrees C). Grease a large baking sheet.
Bring water and amaranth to a boil in a small saucepan. Reduce heat to low, cover, and simmer until amaranth is tender and water has been absorbed, about 10 minutes. Remove saucepan from heat and cool amaranth to room temperature.
Whisk flour, baking powder, baking soda, and salt together in a bowl.
Beat sugar and butter together in a bowl using an electric mixer until smooth and creamy. Beat 1 egg into creamed butter mixture until smoothly incorporated; repeat with second egg. Add amaranth, oats, ginger, and walnuts to creamed butter mixture and beat until mixed.
Stir flour mixture into creamed butter mixture until dough is just combined. Drop dough by the teaspoonful onto the prepared baking sheet.
Bake in the preheated oven until cookies are lightly browned on top, about 10 minutes. Remove baking sheet from oven and allow cookies to cool on baking sheet for 5 minutes before transferring to a wire rack to cool completely.