Expired US passport - dual citizenship

When exiting the US should i give the airline my other passport (knowing that i am flying back to my other country of citizenship) or should i give them my US passport?

Give them your other passport. That's what I always do.

Will the self-check in Kiosks detect that my US passport has expired?

Not if you don't use it in the self-check in kiosk. If you do, it will reject the passport because it is expired.

If i use my other passport to exit the US, will the Kiosks detect that i entered the US with my US passport and am not allowed to exit it with another passport?

No. If you flew in with the same airline, they might notice that you checked in for the incoming flight with an expired US passport, but there is no rule against using a different passport to leave.

What are the risks?

It is "unlawful" for a US citizen to leave the US without a valid US passport. There is no penalty for doing so, however.

I have fairly extensive experience leaving the US on European airlines with an EU passport after having entered with a US passport. Nobody ever mentioned that I hadn't used the EU passport to enter. Nobody asked about my immigration status (though it's a VWP-eligible passport).

There is an extremely small chance that you might encounter a CBP officer screening departing passengers. Such an officer might determine that you are a US citizen who is leaving without a valid passport and prevent you from boarding the plane. (I've never encountered such a screening myself. They are uncommon in general, but I have read that they are fairly common for flights to certain destinations.)

More likely, a US airline might be more interested in the immigration status of departing foreigners than are European airlines. I've read a comment on this site from someone with dual nationality who was challenged when checking in for a flight leaving the US using a passport other than his US one; this was, however, with a US airline, so that is likely why his experience was different from mine. Also, an automated kiosk once required my wife to scan her US visa when checking in for a Delta flight leaving the US. (Her passport was not from a VWP-eligible country; I was using my US passport so I don't know what would have happened had I tried to use my other one.) The airline might have a policy of requiring US citizens who check in with a non-US passport to have a valid US passport on departure. I have never encountered such a policy, however.

When you leave the US, the airlines is primarily concerned about you being allowed into your target country. If you have a valid passport to enter that country, you are good. If you have a round-trip out-and-back-to-the-US (which you don't), they are a bit concerned about how you will get back; they don't want you stranded somewhere and it's possibly considered their fault.

I have been flying many many times to Europe with an European passport, and nobody cared about seeing my US passport (how would they even know I have one?). There is nothing to be concerned about.

Your only concern is that some day or later you need to get a new US passport. Work with the US embassy in your European country, or submit your paperwork while in the US and have it mailed to a trusted friend.

Let's clarify the process a little bit.

In most cases, each international flight involves TWO transactions: departing country A and entering country B. You do NOT need to use the same passport for each transaction. Let's look at US Germany round trip, that I did last week.

1. US Exit: US has no exit controls, so nothing happens here. In the unlikely event someone actually asks, I would have to use my US passport.
2. Germany Entry: German passport. Tip: Always check in with your "entry" passport, not the "exit" passport
3. Germany Exit: German passport. Tip: Always use the same passport for exit that you used at entry.
4. US Entry: US passport. That's also the passport used for check-in since the US is my "entry" country.

So on the return flight to the US I went through passport control twice used both passports: The German one to exit Germany and the US one to enter the US.

(i know about the exemptions to enter the US with an expired passport)

Are you sure about this? The state department website says the exemption expired on Mar 31, 2022. See https://www.state.gov/extension-of-temporary-measure-allowing-return-travel-to-the-united-states-on-expired-u-s-passport/#:~:text=U.S.%20citizens%20currently%20abroad%20whose,States%20until%20March%2031%2C%202022.

When exiting the US should i give the airline my other passport (knowing that i am flying back to my other country of citizenship) or should i give them my US passport ?

No passport needed to exit the US. During check in you give them the European passport, but that's only for entry into Europe not or exiting the US.

Will the self-check in Kiosks detect that my US passport has expired ?

Quite likely. You need to pre-clear travel with an expired passport with the airlines. I would STRONGLY recommend to confirm with them that they will let you board with an expired passport.

If i use my other passport to exit the US, will the Kiosks detect that i entered the US with my US passport and am not allowed to exit it with another passport ?

That's a non issue. See process description above.

Make sure you give the airlines both passports when departing from your other country of origin. This way the airline won't ask for your US passport when you leave the US.

I was able to enter the US with my expired passport, I was stopped for some time and was asked general questions.

I hope this helps.