In a large, heavy-bottom saucepan, combine carrots and 1/2 cup water. Bring to a simmer over low heat; cover and cook slowly until soft, about 30 minutes.
With a potato masher, mash the carrots and any remaining liquid. Stir in sugar, orange juice and lemon juice. Return to a simmer over low heat and cook, stirring constantly, until a thick paste forms, 50 to 60 minutes. Stir in ginger and cinnamon. Let stand until cool enough to handle.
Sprinkle a small cutting board and rolling pin with water. Turn carrot mixture onto the board and roll into a 1/2-inch thick rectangle. Sprinkle generously with sugar. Let stand, covered with cheesecloth or a towel, for one day, until the top feels dry.
Cut into 1-inch wide strips. Turn and sprinkle the other side of the strips generously with sugar. Let stand, covered with cheesecloth or a towel, for another day.
Turn the strips daily, until the candy is dry, up to 4 days. After the strips feel crisp, cut into 1-inch squares. Sprinkle again with sugar. (Store in an airtight container for up to 1 month.)