Combine water and amaranth together in a saucepan; bring to a boil, reduce heat to medium-low, and simmer until water is absorbed and amaranth grains are fluffy, about 20 minutes. Cool for 10 to 15 minutes.

Preheat oven to 375 degrees F (190 degrees C).

Beat cooked amaranth and peanut butter together in a large bowl until smooth. Beat in sugar until completely dissolved. Add butter to amaranth mixture and mix; stir in egg until smooth.

Whisk all-purpose flour, pastry flour, oats, baking powder, baking soda, and salt together in a bowl. Gradually add flour mixture to amaranth mixture, stirring until the dough starts to come away from the sides of the bowl.

Form dough into small balls and place 2-inches apart on a baking sheet. Dip a fork into flour and press dough balls with fork to flatten.

Bake in the preheated oven until set, about 15 minutes. Cool on the baking sheet for 3 to 5 minutes before removing to a wire rack to cool completely.