Upvote:4
You can apply for an ESTA on your British passport, but whatever you do, do not declare your US citizenship on the form when asked about dual nationality, or the ESTA will be refused.
Then, at check-in, present your British passport only, as with an expired US passport, the airline will not let you board. This is because according to the database they use, Timatic, US passports must be valid on arrival in the US.
However, once in front of US immigration, present your US passport. Expect to be taken to secondary and possibly be sternly lectured for not having a valid US passport (which is illegal), and the CBP may want to see your British passport to establish how you got on the plane.
The legal way to go about this would be to get a transportation letter or emergency/temporary passport from the US embassy/consulate.
All in all, to clarify, in practice this trip is doable for you with your UK and expired US passport