In a bowl, with an electric mixer on medium speed, beat oil and brown sugar until smooth. Beat in eggs, vanilla, and applesauce until well blended, scraping down sides of bowl as needed.
In another bowl, mix flour, ginger, baking soda, nutmeg, cinnamon, and salt. Stir or beat into brown sugar mixture until well blended. Let dough stand at room temperature for 30 minutes.
Shape dough into 1/4-cup balls and set 2 1/2 inches apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake in a 350° oven just until cookies begin to brown, 8 to 10 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. Using a wide spatula, transfer the cookies to racks to cool completely.