Help with chickweed

Hi, I have an area of my yard that has significant chickweed growing.
I've been on a organic program for 6 years now.
Lost the grass due to a large water slide for my sons b-day, turned yard into giant mud-pit (kids had fun , followed by chickweed infestation.

Other than hand-pulling (yikes, lot of area to cover), any other methods to control this beast?

Soil test came back with a just under 6 Ph - hit it with lime in fall/spring.

Any help / ideas greatly welcomed!!

Thank you.

Borrow some chickens.
Not kidding. People want their babies looked after during vacations and the sort. Its a nice income, and you benefit for 'weed' control, mowing, and fertilizer. And monies and eggs.
And their poo will raise your ph. Funny how a weed chix love, fixes the problem.

Nature is amazing, low ph, hey lets attract some animals that poo us back to normal.

Interesting idea with chickens. Will the chick eat the grass too?

If I pull the larger mats of chickweed, can my grass out-pace the chickweed and eventually smother it?

I mow high (3.5") and water infrequently (like never).

I guess it depends how willing you are, and how big of a lawn your talking about. manually, rake the matted weeds to raise the runners and try then mowing the lawn short. At the very least the exposed runners will make it easier to pull the weed. Chickweed likes wet and cool, so keep your soil a little on the dry side, it's also better for the grass.  Overseed thinning grass and new bare spots to avoid chickweed infestations and keep the lawn thriving. And then mow the lawn as high as possible to allow the grass to out compete and crowd out any emerging weed seedlings.