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4E width New Balance that I buy when I go the US - they absolutely refuse to sell them here. Although some of the Merril options are pretty wide too. Some of the newer "woven" style shoes work ok as they have enough give - Brook and Asics in particular - but again I only find them in the US. For cycling shoes, Lake. The only people that do really, really wide cycling shoes for cleats. I would love MTB style flats but I simply cannot find anything that fits, but my SPD use goes back to the mid 90s and it's so second nature it doesn't really matter.
My lottery win business would be a shoe company that only makes and sells extra wide casual shoes.
NB, Sketchers, Clarks - Quite s few options for you at 10EE - look for G width shoes or just Wide. Those are usually EE/G width.
If you want a really impossible task, step into my 12 1/2 EEEE shoes.