Heat a 2-quart saucepan over medium-high heat for 2 minutes. Flick water into the pot; if it immediately forms balls and evaporates, the pot is hot enough. If it does not, continue to heat, checking on the temperature every 30 seconds or so, until it is hot. It could take up to 5 minutes to heat the pan, depending on your stovetop. Add about a tablespoon of the whole amaranth and cover the pan. It should immediately begin to pop. Cook, shaking the pan constantly, for 5 to 10 seconds to pop the grains. Remove from the heat, give the pan a few more shakes, then uncover and pour into a medium bowl. Wipe out the saucepan. Continue popping the remaining amaranth one tablespoon at a time. When finished, let cool while you make the dough.
Whisk the amaranth flour, dried orange peel, salt, and baking powder together in a small bowl. Set aside.
Place the brown sugar and butter together in the work bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed, scraping down the sides and bottom of the bowl as needed, until light and fluffy, 3 to 5 minutes. Still on medium speed, beat in the egg until smooth, about 30 seconds, followed by the vanilla, about 15 seconds. Scrape down the sides and bottom of the bowl. Return the mixer to low, then slowly add the flour mixture, followed by the popped amaranth and candied orange peel. Continue to beat until combined, 10 to 15 seconds. Remove the bowl from the mixer, cover, and chill for 30 minutes.
Meanwhile, heat the oven to 325°F with a rack in the middle of the oven. Line two half sheet pans with parchment paper.
Drop the batter by the tablespoonful onto the prepared pans and parchment, 12 to a pan, 2 inches apart. Gently press down on the dough rounds to form cookies about 1/2-inch thick. Bake the cookies, one pan at a time, until the cookies have puffed, formed a cracked top, and are golden brown on the bottom, 10 to 12 minutes. Cool for 1 minute on the pan before moving to a wire rack to cool completely. Repeat with the second prepared pan, then re-line the original pan to bake off a third batch. Sealed in an airtight vessel, the cookies will keep for two weeks. They freeze well, too.