Arrange a rack in the middle of the oven and heat to 350°F. Coat a 12-cup muffin tin with oil or line with paper muffin liners.
Stir the olive oil, applesauce, sugar, cinnamon, ginger, and nutmeg together well in a large bowl. Add the carrot and water and stir to combine. Add the flour, oats, raisins or dates, baking powder, baking soda, and salt. Fold these dry ingredients into the wet ingredients until there are no more floury patches left.
Divide the batter between the wells in the muffin tin. Bake until a toothpick inserted into the center of a muffin comes out clean, 25 to 30 minutes. Let the muffins cool in the tin for 15 minutes — they may sink a bit as they cool. Transfer the muffins to a wire rack and cool completely.
