I'm looking for a type of chicken that I can use in a chicken yard to keep my grass mowed, will grow quickly, stay in their yard to clean all of the grass out. What breeds would you recommend that I get for my lawn mowers?
You're confusing concepts. Chickens are used to prepare the ground prior to the planting of the food forest. They weed the ground, fertilize and lay eggs. They are confined inside moveable fencing, and any domestic breed will do the job. In no universe do they mow lawns.
FORGET chickens, bucko. They might eat seeds and seedlings but in no universe I know they actually cut grass down. My goodness. If you are wanting A LAWN forget about hiring goats, horses, cattle...sheeps. That is fine for pasture but you'd have to be able to rotate them before they eat down to the crowns and demolish any grass plants you have.
If you want a lawn just get your man juices up and purchase a damn good hydraulic mower with blades you can sharpen, bag your grass clippings, purchase a cheapo scott's hand spreader for fertilizer and/or seed, definitely need a great gas powered string trimmer and a gas powered blower (Stihl is a bonafide great brand for homeowner use...). Otherwise, make an lawn of gravel...like the Japanese do. Once per year rent a plug aerator...
Chickens are not lawn mowers. They eat grass, but they select what they like (plants with relatively large leaves are preferred, like clovers, Plantago, etc). So you have no uniform cutting.
They like also much the ground, so they will start to throw away grass in patches, so that they can delouse (if it is dry) or looking for worms (if is is more humid). In a forest, I would expect they will dig various holes (not deep, but large).
But if you want a wild garden, they could help, and also in a wild garden (OTOH a wild garden doesn't requires chicken nor lawn mower).