You might just need to roll them thinner and/or bake them at a lower temperature for a longer period of time, so they cook through to the center before the edges get too brown. Also baking on a silicone mat will even the heat distribution to the cookie a lot, and lead to more even baking/browning. But some browning of the edges is likely inevitable with a cookie this large.
Both my sugar cookie and gingerbread recipes bake fully through to the center at this size with no trouble; I really don't alter anything with the recipe, only keep them in the oven slightly longer.