OK, I'll be the one to disagree about plug size.
For starters, a "plug" should NOT be cut. A flap is much better. Grass has what is known as a "rizome" root system, which means that several plants shoot up along an extended root. Each shoot is able to draw a certain amount of support from neighbors along the same parent root. Separating a "plug" from the parent roots will almost assuredly kill the plug in hot dry conditions, whereas leaving it attached along at least one side will allow for neighbors to help keep it alive.
If you're only going after shallow targets, a slit or "squirrel dig" just through a small spot in the surface won't be noticed along side the mower scalping and mole holes/tunnels and other squirrel diggings. But, if you're chasing deeper targets, a larger deeper initial hole will upset the root system the least and do the least lasting damage.
And lastly, small or shallow plugs and flaps are much more likely to be sucked up, chopped up, and spread accross the landscape by a 20+hp zero-turn mower than a larger, deeper, heavier flap.
If you're hunting somewhere that is sensitive in dry conditions, then there is no substitute for carrying a couple gallon jugs of water in the car. One can be carried while hunting and used to water in the areas that you've disturbed. The worst part of doing this, is that sometimes the place that you've dug and watered is the only place that it stays green when everything else burns up.
FWIW, my experience comes from working 8½yrs for a Big-10 university grounds department.