Luckily I have local friends so I texted one and got 3 recommendations--Russian, Uzbek, and Kazakh. The Russian and Uzbek ones were within walking distance, so we walked.
These two restaurants are located with several more expensive restaurants, in the area that's called the 'round square'.
This round square is located behind the KazMunaiGas building that is across the park from Khan Shatyr. If you're standing in front of Khan Shatyr, you can see Lovers' Park across the street, and behind that the KazMunaiGas building. You can even see beyond, to Beiterek. It's a lovely view and a lovely walk.
We found the Uzbek restaurant, but did not see the Russian one, Самовар (Samovar), so we ate at the Uzbek one, Кишлак, (Kishlak). As far as I know, Uzbek and Kazakh foods are very similar; at least, this restaurant offered quite a few dishes that I associate with Kazakhstan.
We settled on shashlyk and lagman, with baursak for bread and some vegetables. Shashlyk is like kebab--it's meat grilled on a stick--and the one we got was lamb mixed with something else. Usually the shashlyk I get is not mixed or ground, it's just the meat, and I did not like this one as much.
Lagman is a type of noodle; this one was served in a soup-like mixture with meat. It was very good. Baursak is a puffy fried bread-ball and is one of Sophia's favorite foods.
The restaurant seemed very Uzbek to me, the waiters were in traditional costumes, and the tiles on the walls reminded me of the mausoleum in southern Kazakhstan. My friend, who had spent the past several years in China, said everything--from the décor to the costumes to the food--reminded her of western China!
So I'm happy I got to experience a local restaurant, maybe someday I'll try Samovar or the Kazakh restaurant.