For the syrup: Place the maple syrup and ginger in a small saucepan and bring to a boil over medium heat. Reduce the heat to low and simmer for 10 minutes. Use a slotted spoon to remove the ginger and discard. Stir in the bourbon and keep warm while you make the French toast fingers.
For the French toast: Place two racks in the upper and lower thirds of the oven and preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper and spray lightly with cooking spray.
Slice the brioche into 2-inch slices and then cut each slice into three to four 1-inch-wide fingers. In a large bowl whisk the eggs, milk, butter, vanilla, bourbon and salt. Dunk the fingers into the egg mixture, let the excess drip off, and then place them onto the baking sheet. Bake the fingers until they are firm and golden brown, 15 to 18 minutes, flipping the fingers over halfway through baking. Serve warm with the syrup.