In a food processor, pulse the almonds until you get chopped almonds ranging from finely chopped to mostly finely chopped.
In a stainless steel bowl, whisk the egg whites with a fork until frothy. Stir in the sugar, pumpkin pie spice, salt and cayenne. Add the carrot, coconut, almonds and orange zest and stir until everything is moistened. Set aside for 10 minutes to dissolve the sugar and hydrate the coconut.
Set the bowl directly in a wide skillet of barely simmering water and stir the mixture with a silicone spatula, scraping the bottom to prevent burning, until the mixture is very hot to the touch and any liquid at the bottom of the bowl has thickened and turned from translucent to opaque.
Drop by heaping teaspoons 1 inch apart on parchment lined baking sheets or on silicone baking mats. Bake about 25 minutes, until the tips of the carrot shreds begin to color. Allow to cool completely.
Remove the cookies from the pan as best you can—use Robbie’s trick if you lined your pans with parchment.
Keep in a box loosely covered—they will loose their crunch if kept airtight.
Ms. Medrich’s recipe says do not freeze these cookies.