I'm not sure I can speak to ice cream specifically in Persia 1000 years ago, but the history of ice is fascinating. I'm no historian, but take a casual interest in all things culinary and hospitality.

100 year ago, many homes and inns would have ice houses - in the winter, they'd harvest ice and stack huge blocks in a building, insulated with straw. This would last all summer, allowing people to have ice in a drink. In cities, this involved huge warehouses and teams of icemen. I actually have the remnants of an old ice house on my property.

While not answering the question of how they actually did it, you can make ice cream and other frozen desserts with nothing more than 2 metal bowls, a spatula, and some ice.

Put ice and water into a large bowl, and put a smaller bowl inside that. The smaller bowl will contain your ingredients, but remember that ice melts at 0°C, which means that a big bowl of ice water will never be colder than that, and thus, won't freeze anything else.

The key is adding salt to the water - saltwater melts ice at temperatures as low as -5°C, which is cold enough to freeze a thin layer of fruit juice, or cream, on the side of the bowl. If you keep mixing, you end up with more and more frozen dessert.

Of course, this doesn't allow you to keep the ice cream/granita for any amount of time, so it was probably prepared "a la minute" - in the moment, right before being served.

If it's served at market, I'm gonna take a highly illegal guess and assume they'd work off big blocks of ice and process it on site.

I know that doesn't answer your exact historical question, but I think it's an interesting side note. I hope that's not too far outside the spirit of the subreddit.