Can you identify these large pipe shaped chunks that came out of my cold water tap?
My best guess is limescale that's been knocked free by some kind of work upstream. You could contact your local water authority to see if you might have been affected. 

You're right to clean the aerator screens in your faucets, etc. The only other thing you might consider (after running water for quite a while until it turns up clear) is operating all the shutoffs in the house. Open and close them a few times.

(Pro-tip: know where your main shutoff is and know that it works before you mess with the local shutoffs.) 

Operating the shutoffs might chew up the washers and prevent the shutoff from working, so be prepared for this possibility. I mention this step because I'm of the 'it's better to know now and fix it, rather than when it's an emergency and you have water pouring into the sink cabinet' school of thought.