I've used this place once:
https://iqbirdtesting.com
They've got a test for a blue egg gene.
They keep re-arranging their site, but right now I can find it listed on this page:
https://iqbirdtesting.com/order-bird-dna-test/
Scroll down and look for pictures of blue eggs

It's currently listed three different ways, according to what kind of sample you send in (blood sample, or a few plucked feathers, or the shell that a chick hatched out of. Obviously you don't have the eggshell for your cockerel, so it would be blood or feathers.)
I've also looked at the website for this place:
https://thesilkielab.com/
This is currently the page with the blue egg gene test:
https://thesilkielab.com/product/blue-egg-gene/
I have no personal experience with this company, but it has also been some years since I used the other company, so I'm not really in a position to give a recommendation about which one would be better.
For either one, you go to the website and follow the instructions. It's probably something like place an order, set up an account, collect samples, mail the samples, wait a bit, get the results through the online account. But trust the company's website on the exact details, not me.