You would need to use a text field instead of a select field and use jQuery autocomplete(http://jqueryui.com/autocomplete/).
There's a nice example on how to use it.
You would need just to populate a variable in js with all the breeds in it and then use the plugin to filter the list and use one of its callback in case you need to customize your text field further; e.g. after a user has chosen the breed.
Hope this helps you