Position racks in the upper and lower thirds of the oven; preheat to 400 degrees F. Line 2 baking sheets with parchment paper.
Sift the cake flour onto a sheet of parchment paper and set aside.
Combine the whole egg, egg white, sugar, baking powder, salt, vanilla extract and water in the bowl of a stand mixer fitted with the whisk attachment. Beat on high speed until lightened in color and thickened, 3 to 4 minutes.
With the mixer running on low speed, slowly pour in the butter and oil. Increase the speed to high and beat for 30 seconds.
Turn off the machine and add all of the flour. Mix on low speed. Stop to scrape the bowl and whisk. Mix on low speed until the batter is smooth and thick.
Transfer the batter to an 18-inch (or larger) pastry bag fitted with a coupler and 1/2-inch round tip. Alternatively, use a gallon-size resealable plastic bag with a corner snipped off to make a 1/2-inch opening.
Pipe 5 quarter-size mounds of batter in 8 rows on each of the prepared baking sheets. You should have 80 mounds total. Pipe any extra batter into the empty spaces.
Bake, rotating and switching the positions of the baking sheets halfway through, until the edges of the wafers turn a deep golden brown, 14 to 15 minutes total.
Let the wafers cool completely on the baking sheets. They will crisp as they cool.
The wafers will crisp as they cool on the baking sheets. Should you find that they haven’t become completely crisp in the center, you can rebake the cooled wafers at 350 degrees F for 5 to 6 minutes.