When our kids were that age we went to David Chen in Armonk quite a bit. They have a downstairs room where they tolerate kids walking around, and there is a small area where kids can lie on the floor and watch a video that's always playing. The food is probably a little better than the normal Chinese around here, but nothing special.
I know you said Stamford, but this is probably 20 minutes away if you go the backroads.
I take my little kids to Ole Mole quite a lot. Fine for kids, though I can't recall if they have high chairs. But it sounds like your kids are currently in that in between stage anyhow, so shouldn't be an issue.
There isn't a lot of space in there, but that's really only impacted me when I bring all four of my little tykes.