My EU passport was stolen and I am going on holiday to Thailand from the UK this week. I have dual nationality and a US passport, can I use this to leave and re enter the UK?
You can certainly do this, but you may have trouble entering the UK at the end of your trip, because you will either have to lie about your intentions in the UK, which is a terrible idea, or convince them through other means that you are an Irish citizen and therefore entitled to enter and reside in the UK.
The specific law that allows you to enter as an Irish citizen without an Irish passport is regulation 11(4) of the Immigration (European Economic Area) Regulations 2016, which says
(4) Before an immigration officer refuses admission to the United Kingdom to a person under this regulation because the person does not produce on arrival a document mentioned in paragraph (1) or (2), the immigration officer must provide every reasonable opportunity for the document to be obtained by, or brought to, the person or allow the person to prove by other means that the person is—
(a) an EEA national;
It will help to have a credible explanation of why you do not have your Irish passport with you, so do bring the police report showing that your passport was stolen. You should probably bring as much additional evidence of your Irish citizenship as you can possibly get your hands on before you leave. The police report might be enough, but more evidence can only help.
Also see the related question Immigration officer stamped “Indefinite leave to enter UK” in US passport of EU citizen. How to fix it?, in which a Swedish/US dual national was admitted on the basis of Swedish nationality (albeit with an incorrect stamp) after presenting a US passport; in that case, the traveler didn't even mention the EU nationality until after the officer asked.
Option 1: If you have entered the EU using US passport than you can also leave the EU using US passport and no one will ask you a question.
Option 2: But in your case you are living permanently in UK (and I think your residency is based on EU passport and you have entered the EU with your Irish passport), you have to file a police complaint about the lost passport (and request the police report if possible). Then you can apply new passport using "Online Passport Renewal Service" or go to the nearest Irish Embassy or Consulate. There you have to wait a minimum of eight working days. But with valid proof of confirmed travel (e.g. airline ticket/email) the application can be processed sooner with additional fee. Get the time slot as soon as possible