Nothing is impossible. You just need to do some work.
1. Generate standard static page using PHP or smth else. The page should look like the page of your ExtJS app. But all links must have GET params in URL. Also PHP should aggregate input GET params.
2. Add your ExtJS app to the page. In the app you have to take into an account GET params and make proper request.
2a. If a real user opens your page: PHP generates the output, then ExtJS app starts and hides the static page and generates the dynamic output.
2b. If a crawler opens your page so JS is disabled, PHP aggregates the request according to GET params and generates the output.
You can add params to URL like #param1¶m2¶m3 in ExtJS when clicking on links, so real users will be able to share their links. Just learn the router on PHP-side to understand URLs like this.
There is no way to make SEO-friendly pages using JavaScript only.