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.

This pioneer-era pie is due for a comeback. If you were to flip open a pre-1975 edition of The Joy of Cooking, you might be startled by an illustration that demonstrates how to skin a squirrel. The ...