Like countless Americans, Kimberly Darling celebrates Thanksgiving with a bountiful, home-cooked feast. Her take on the holiday has a swampy twist: She forgoes the familiar turkey in favor of an alligator she traps on one of her many hunting expeditions, then she brines, smokes and wraps it in bacon before serving her guests.

“People walk in and they’re like, ‘Oh, my God, that’s a literal alligator. What did we sign up for?’” she says.

But...