Cut each slice of thick bread into three even vertical strips. Using slightly stale bread works best as it absorbs the custard without falling apart.
In a wide, shallow bowl, whisk together the apple cider, eggs, heavy cream, brown sugar, vanilla extract, one teaspoon of cinnamon, and nutmeg. Whisk until the eggs are fully incorporated and the mixture is smooth.
In a separate small bowl or shallow plate, combine the granulated sugar and the remaining tablespoon of cinnamon. Mix well and set this aside for the final coating step.
Place several bread sticks into the cider custard. Let them soak for about 15-20 seconds per side. You want them to be well-saturated but not so soggy that they break when lifted.
Melt 2 tablespoons of butter in a large non-stick skillet or griddle over medium heat. Place the soaked sticks in the pan, leaving space between them. Cook for 3-4 minutes per side, or until they are golden brown and slightly crisp on the edges.
Immediately after removing the sticks from the skillet, toss them in the cinnamon-sugar mixture while they are still hot. The residual heat and butter will help the sugar stick to the surface, creating that signature donut texture.