I'd argue that she _doesn't_ grow food in hard clay, but instead has turned that hard clay into soil with years of work.
It’s amazing how much damage people do to their lawns by consistently over-mowing.
As a new homeowner, I'm curious: what would you define to be "over-mowing"?
That is, mowing even if the grass doesn't need it. For example, the lawn service comes once a week, always on the same day.
It's also going to depend on your mower. Is it push electric, push gas, or ride gas? The more power you have the longer you can let it grow. And the longer you can let it grow - within neighborly reason - the better.
I like to cut at maximum height on the typical homeowner mower, or at most one notch below, and every two weeks or less. Spring is the only time when I end up cutting every week due to peak growth rate.
I don't bother with herbicides but add fertilizer once a year. There is a lot of weeds that don't belong but as long as they are cut they don't bother me. I also like the dandelion colors in the grass, and the kids like the fluffball seeds. If you are not OK with weeds you'll have your work cut out for you - there will be an order of magnitude more steps involved.
Some grasses will never grow to maximum height of most mowers, as a general rule, cut it so that you're leaving more than enough of the grass uncut that its still getting good ground coverage and there the 'lawn' is not being destroyed during the mowing process.
The first time I mowed, it had been growing for about 4 months. I ended up cutting close to three feet of grass and cut it down to about 6 inches -- my mower's max height. The next week I cut it to two inches. Then with winter it didn't really grow for another three months or so. Now I'll cut it to four inches and it's a foot long in two weeks.
I've got five acres and about four of it is lawn with another acre of trees. I've got a strong powerful mower. Weeds don't bother me as much as taking four hours to mow. I knew it would take a while though and I'm perfectly fine with letting the grass grow long for three or four weeks between mowings. But finding how much cutting is too much will take a while for me.
I would let it grow all year, and never mow, if it weren't for the pests that attracts. Neighbors don't bother me. In fact, neighbors are interested in leasing for their cattle. That's the great part of having so much land. But I'd rather not step on cow pies ;) I'd be more interested in letting someone grow crops for a quarter of the year. I don't have time to do that myself.
You might be able to partner with a local school or scout troop to organize a community garden, or an educational garden.
The “improve the ground” is a very important part of subsistence and sustainable farming. Reducing the outside inputs as much as possible while still improving the soil is a laudable goal.
But tilling it at the beginning to create ideal starting conditions for such a no-till approach is allowable and encouraged.