Butter a 9 by 9-inch baking dish and line it with plastic wrap going up the walls of the pan.
Cut up the croissants into 1-inch cubes and place them on a sheet pan. Place in oven and toast for 10 to 15 minutes until slightly crisp, then place them in a bowl.
Meanwhile, in a saucepan, heat the half-and-half, cream, salt, and vanilla bean over medium heat, stirring occasionally to make sure the mixture doesn't burn or stick to the bottom of the pan. When the cream mixture reaches a fast simmer (do not let it boil), turn off the heat. Set aside to infuse for 10 minutes.
In a large mixing bowl, whisk the eggs and sugar together. Whisking constantly, gradually add the hot cream mixture. Strain over the croissant cubes to remove the vanilla bean, and let soak 15 minutes, gently turning the cubes over once. Empty the mixture into the baking dish. Let soak another 30 minutes, or overnight, in the refrigerator. Heat the oven to 350 degrees F.
Place the baking dish in a hot water bath. Bake until just set and very light golden brown on top, about 25 to 30 minutes. Let cool, then chill in the refrigerator 4 hours or overnight. The next day, turn out the pudding and cut round 3-inch plugs from the pudding, place them on a microwave-proof pan and cover with plastic wrap.
To serve, warm maple syrup in a saucepan and keep warm. Warm the plugs of bread pudding in the microwave and serve topped with hot maple syrup. Serve with ice cream.