In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg.
In a separate large bowl, whisk together the olive oil, granulated sugar, and brown sugar. Whisk in the eggs one at a time. Add the sourdough starter, 1/4 cup of the caramel sauce (save the remainder for the cake layering and frosting), milk, and vanilla extract. Whisk until smooth and fully combined.
Fold the dry ingredients into the wet ingredients until just combined. Be careful not to overmix.
Gently fold in the shredded carrots.