As the kitchen manager at the Odd Soul, a pizzeria and bar in Springdale, Arkansas, Addie Hayes is used to spending Saturday mornings in front of a stove. But today, she’s a little out of her element.

Young fryer squirrels are always a treat. Coated and cooked golden brown, they’re morsels few hunters will ever pass up. But what about those old ones? You know, the squirrels that were probably ...