Mix flour, baking soda, ginger, cinnamon, cloves, and salt together in a large bowl.
Combine brown sugar, oil, molasses, egg, and 1/4 teaspoon salt in a large bowl; beat with an electric mixer until smooth. Pour over flour mixture; mix until a dough forms.
Roll dough into 1-inch balls and arrange on ungreased baking sheets.
Bake in the preheated oven until firm, 8 to 9 minutes. Let cool on the baking sheets for 5 minutes. Transfer to a wire rack to cool.
Place shortening in a large microwave-safe bowl. Heat in the microwave until melted, about 1 minute.
Stir confectioners' sugar and orange extract into the melted shortening. Mix in milk a little at a time until glaze runs smoothly off the end of a spoon. Add 1/4 teaspoon salt. Spoon glaze over cooled cookies.
