Preheat oven to 325°.
Combine crystallized ginger and 1 tablespoon flour, tossing to coat; set mixture aside.
Combine the sugar, molasses, cinnamon, ground ginger, and cardamom in a medium saucepan. Bring the mixture to a boil over medium heat, stirring occasionally. Stir in baking soda. (The mixture will become thick and foamy.) Remove mixture from heat. Add the butter, stirring until butter melts. Add the egg, and stir well with a whisk. Lightly spoon 2 2/3 cups flour into dry measuring cups, and level with a knife. Gradually stir 2 2/3 cups flour and crystallized ginger mixture into sugar mixture.
Turn dough out onto a flat surface, and knead lightly 4 to 5 times. Cool slightly. Divide dough in half. Roll each portion into a 1/4-inch thickness on a lightly floured surface; cut dough with a 2 1/2-inch cutter. Place on baking sheets coated with cooking spray. Bake at 325° for 12 minutes. Cool the cookies on pans for 2 minutes or until firm. Remove cookies from pans; cool on wire racks. Repeat the procedure with remaining dough.