From Tash Rabat, roughly six hours on foot or by plodding horse should get you to a broad ridge overlooking massive Chatyr-Köl (Чатыр-көл) – 23km long and 11km at the widest, but only around 20m deep. Continue for a couple of hours and you can stay the night in a yurt on the shore at just over 3500m, but note that after rains the area is often a boggy mess. The yurt owners at Tash Rabat can arrange horses and guides.
Visiting the area requires a border-zone permit, though yurt owners report that these are rarely checked for hikers approaching over the mountains. However you'll need to return the same way as these are certain to be checked at a police post on the road route, and ideally all visitors would prepare one before visiting.