This probably literally means "The dog's breakfast" eg, what the dog is eating. But there is a phrase in English "The dog's breakfast" which means a much of random things thrown together. It could be food. But it could also be a bunch of different jobs you have to do or something.
I know we're learning Greek here, but let me indulge myself in a bit of English for a moment. We've got all three forms of the article:
A dog's breakfast as in It's a dog's breakfast: Brexit is a real dog's breakfast (as Tom says and who will disagree—stand up and be counted!)
The dog's breakfast as in: Who ate the dog's breakfast?
And zero article: I never eat breakfast.
Please don't expect me to explain, though I will if anyone really wishes.
This thread started two years ago, before Brexit was even a twinkle in Boris Johnson's bloodshot eye. :-)