I dunno, but I'd never associate this with hamburgers. Its origins are Turkish, although so much changed that turks would never recognize it.

Traditionally, the meat is of several sorts (beef, pork, sheep - never lamb, and, if available, about 1/8 of the total meat should be horse or donkey - slaughtered of old age, not specifically for the meat, young meat of these animals doesn't contribute the right way to the flavor).

Also traditionally, in order to obtain a nicer texture, you should chop the meat, not grind it. Grinding presses the juices out, and leaves the meat with a saucy texture, but that's not the real thing anymore.

Allspice ... cannot be. This must be a modern and foreign addition in your recipe.

You should leave the meat mixture overnight in the fridge to have the spices' flavors invade the meat. You can afterwards taste the difference.
This can be a healthy dish. Broiling them in oil turns them into an unhealthy one. So you should never broil them, and not grill them on the highest setting. And definitely not until they are crisp, but just until they feel like a rare or medium done steak. Being minced or chopped meat, they're close to being burnt if you grill them until they're crisp. By loosing fat, they become somewhat spongy during grilling, and need not be feeling like well done before they are really done.

Also, try not to turn them that often - turning them once only is ideal. You risk breaking them and spreading the minced/chopped meat all over the grill. When you place them on the grill, they will instantly stick to it. They are rare the moment they un-stick themselves (due to the melting fat), so you shouldn't turn them before that.