Its Emmerdale.
I'm from Cumbria, not Yorkshire so I'm not sure of my ground here.
The Dingles call people 'fat egg' when they're accusing someone of being daft or thick. Around here people say 'fat head' ...... like they might say "you should have taken an umbrella you fat head" when someone's been caught in the rain.
If they're being a bit less polite they might say "you should have taken an umbrella you dick head" .... but I've never heard fat '*egg*'.
I wondered of it was a Yorkshire thing?
A few episodes ago, when Cain was teaching Sam how to fight, so that he was ready for life in prison, he used the term to goad him on, and they both said it back and forwards to each other. I remembered hearing them use it a few more episodes ago too.
Light hearted comments only please, this is not intended to poke fun at different accents.