Cut the bread into 1-inch slices, then cut each slice into three sticks. Slightly stale bread works better here than fresh.
Whisk the eggs, milk, vanilla, 1 1/2 teaspoons cinnamon, and salt in a shallow dish.
Dip each stick in the custard and turn it once. Two or three seconds per side is plenty. Bread left to soak turns to mush in the middle.
Melt some butter on a griddle over medium heat and cook the sticks 2 to 3 minutes per side, turning to brown all four sides.
Stir the rolling sugar and remaining cinnamon together, roll the hot sticks in it, and serve with maple syrup for dipping.