A number of places that serve pizza for lunch/dinner also offer it for breakfast and brunch, usually with an egg or two on top. Here's a couple in the East Bay:
Nizza la Bella has two egg pizzas, the Cyclops (with potato and sausage, no tomato sauce) and the Bonjour Bella (a basic cheese pie with egg). I like the Bonjour Bella, personally. These are just the right size for one person, and only around $8.50.
Zut! in the Fourth Street complex usually has a pie with spicy sausage, fresh mozzarella, two eggs and a sprinkling of arugula. It's $15, but can easily serve two (I bring half home for dinner).
I've seen egg-topped pizza at PIQ in downtown Berkeley but haven't tried it.
Rotten City experimented with an egg pizza on weekends, but they haven't mentioned it on Facebook in quite a while.
The cafe at Olivetto has an egg pizza. I've wanted to try it, but for a long time breakfast hours ended at 11:00. I see from their website that the brunch menu continues into the afternoon now. $16.
And, two others in passing: the Prosciutto pizza at Paisan would be incredible with a couple of eggs on it. When I mentioned it to a waitress, she said people asked for it all the time, but the restaurant doesn't seem to be interested. And the "carbonara" pizza at Bucci would make a great breakfast, but they're not open then.
Over in the City, I see that Zero Zero has a brunch pizza, haven't tried it.
What are your favorites?