Preheat and prep. Preheat oven to 350°F (180°C). Line a baking sheet with parchment paper and lightly grease with cooking spray or a thin brush of melted butter.
Cut the bread. Slice each piece of bread lengthwise into 3 even sticks, about 1 inch wide. Set aside.
Make the cinnamon sugar. In a small bowl, combine ⅓ cup sugar and 1½ tsp cinnamon. Set aside for coating.
Make the egg batter. In a shallow dish, whisk eggs, milk, heavy cream, vanilla, 1 tbsp sugar, 1 tsp cinnamon, nutmeg, and salt until fully combined and smooth.
Dip the sticks. Quickly dip each bread stick into the egg batter, coating all sides. Do not soak — a quick turn is enough. Let excess batter drip off before placing on the baking sheet. Space sticks apart so they don't touch.
First bake. Bake on the center rack 10 minutes until the bottoms are set and starting to turn golden.
Flip and finish. Remove from oven; flip each stick with tongs. Return to oven and bake 6–10 more minutes until firm, golden brown, and crisp on all sides.
Butter and coat. Transfer sticks to a wire rack. Immediately brush all sides with melted butter, then toss or roll each stick in the cinnamon sugar coating until fully covered.
Serve warm with maple syrup for dipping.