When I scrape out the seeds out of my vanilla beans and simmer them with milk or incorporate them into batter, they always leave little tell-tale black dots in the finished product. Is it possible to filter them out somehow and is it something commonly done in the professional kitchen? I don't think I remember seeing these dots in restaurant desserts. It's not critical, but I'd rather my panna cotta be pristine white :-).
A chinoise will not--I speak from daily experience--filter out vanilla seeds. A coffee filter would, but my question is: why on earth would you bother? It's annoying to do (and depending on what the product is, filtering through a coffee filter would ensure you lose other things), and seeing actual vanilla seeds in the final product tells your guests that you are using real vanilla beans, not extract.
Professional kitchens do not, generally speaking, attempt to remove the seeds for the reasons noted above.
@stephenmacdonald, most dessert recipes that call for a fine mesh strainer do so because there is the possibility of chunky things ending up in the product. In the case of that recipe, I would guess it is to prevent chunks of undissolved gelatine from ending up in the final dessert.
Just use vanilla extract if you don't want specks of seeds showing. Straining of custards is to remove any coagulated egg particles, not to remove the seeds.
infuse the cream with the whole bean intact, do not split it. It will still infuse flavor, not as quickly or as much but that would be one way to use it without specks showing in your finished dessert.
A coffee filter should do the trick. I believe cheesecloth wouldn't be fine enough, but I'm not 100% sure.
This recipe for panna cotta calls for scraping seeds and using a fine mesh strainer on the final product, but as per @roux and @Darin's answers, it's not for getting the seeds out.
I'm not certain as to whether professional kitchens strain them out.