Food producing grass areas?

Any ideas or experiences with having an area of grass or "lawn" that produces some kind of food, yet it's nice to walk or play on like a traditional grass yard?  I'm thinking about a climate that has 4 seasons, not subtropical, tropical, etc.  My family all would like to have some kind of lawn space for games or just running around barefoot, so I'm trying to think of a way to make it productive or useful rather than just the common resource intensive monocrop lawn seen around most houses in the U.S.

Just let the weeds grow.

Seriously common lawn weeds like dandelions and daisies are edible as is clover and sorrel.  If you keep it mown pretty short thyme and camomile will also thrive.

There are many grass alternatives- however, they get trampled you wont eat them as much.  Best to include a few fruit trees or bushes along the edge and then create a lawn alternative  that companions well with these fruit plants.  Perhaps, a mix of creeping thymes, clovers and with chives on the edge.  You can always mow these or let them sprawl depending on your goal.  But it low mow anyways.