In a medium bowl, whisk together eggs, milk, vanilla extract, ground cinnamon, and nutmeg until well combined.
Cut the bread into quarters and dip each piece into the egg mixture, allowing it to soak for a few seconds.
Heat a skillet over medium heat and melt 2 tablespoons of butter. Cook the soaked bread pieces for 2-3 minutes on each side until golden brown.
In a small saucepan, melt the remaining butter and stir in brown sugar until dissolved. Drizzle this mixture over the cooked bites and toss gently to coat.
Dust with powdered sugar and serve warm with maple syrup.