The simple answer is no, because unlike when ordering eggs or other things that have various levels of cooking methods I have never once been asked when ordering hashbrowns how I want them done.
Of course you can still try, but the biggest problem with what you are asking is that it's not straight-forward to make them that way -- and let's be honest, most places you get hashbrowns from aren't the type of places that can handle complicated or unusual requests.
"crispy to the point of dried out or even burnt.
I'll disagree with this a bit -- there are 2 styles, what you describe but also generally undercooked overall. And this is where the problem lies, at your typical breakfast food diner the cooks are using flat tops with extremely limited temperature control. What you are asking for is akin to a rare steak cooked at a good steakhouse; it has a well developed crusty exterior with barely any interior cooking.
The problem is you are looking for something in between these two and it would require rather specialized cooking (you need an extremely hot cooking surface to get the exterior crisp without drying out the interior) and a good amount of attention to get right. Maybe you can get this at a fancy brunch place. At most breakfast places this simply is not going to happen, they are cooking hashbrowns in batches for multiple orders and it is largely an afterthought while they work on the other dishes.