How to make Espresso with an Aeropress - there are many different opinions on how best to do this. I've tried all of them. The most important concept when it comes to espresso is pressure, and there's only one way to make pressure with an Aeropress.
The method I use for making an 'espresso' with the Aeropress is simple but leaves room for adjustments and experimentation. How long you 'pre-infuse' (or bloom) your coffee will have an impact. Whether you decide to do this inverted or not will have an impact. Whether you use a metal screen or not will have an impact. In the end, the most important thing is that you are using high pressure and a fine grind.
This AeroPress recipe uses a paper filter, but I urge you to try with a metal filter too if you have one.