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In the commercial egg business,
chickens are typically fed just grains, and they continue laying.
Fresh pasture (with its delicious bugs for protein) plus garden scraps will improve the quality of the eggs.
In your climate, you might be able to get away with some late sowings of buckwheat.
Many use it as a green manure crop because it is such a fast grower - 30 days from sow to mow.
Might be able to extend your 'greens' season a little longer with that.
Things like winter squash & pumpkins that aren't storing well make delightful 'toys' for them to eat. Dense & filling.
Rather than making special trips to the store for
chicken feed, how about just stopping by there when you are passing?
Now that the growing season is mostly over, they will probably have less scrap.
Once you get spoiled with fresh, home grown eggs, you will
not want the store bought kind any more.