Preheat oven to 350°F. Spray an 8- or 9-inch cake pan with baking spray and line with parchment paper.
Combine sugar and lemon zest in a large mixing bowl. Mix with a rubber spatula, pressing to bring out the lemon oils.
Add flour, cocoa powder, baking powder, ginger, and salt, whisking after each addition.
Stir in chopped ginger to coat with dry ingredients.
Add olive oil, eggs, milk, and vanilla extract, whisking gently between each addition.
Pour in boiling water and fold to combine completely. Batter will be very liquidy.
Pour batter into your prepared cake pan and bake in preheated oven for approximately 30-35 minutes, or until a toothpick inserted into the center of the cake comes out clean. (see note)
Let cake cool 20 minutes in pan, then transfer to a wire rack to cool completely. Dust cooled cake with powdered sugar, if desired.
Baking time for different size pan: You can make this chocolate olive oil cake in either an 8- or 9-inch cake pan, but if you use a 9-inch pan it will cook a bit faster. I found 35 minutes to be perfect for an 8-inch pan, but if you're using a 9-inch pan, I would check the cake after 27 or 28 minutes.