Remove the banana pieces from the milk using a slotted spoon or skimmer. You will not need the strained banana pieces for the rest of this recipe but you can save them for smoothies or other baking recipes.
Pour 1/2 cup of the infused banana milk into a mixing bowl, then add the honey, egg yolks, salt, and arrowroot powder. Whisk together.
In a medium saucepan, reheat the remaining milk over medium heat, stirring gently until simmering.
Very slowly (just a few tablespoons at a time) drizzle about half of the hot milk into the egg mixture while whisking rapidly to keep the milk from curdling the eggs.
Return the saucepan to the stovetop, pour the tempered egg mixture into to saucepan, and warm over low heat, stirring constantly until the pudding mixture simmers and thickens slightly, about 3-4 minutes. Watch carefully to make sure the pudding mixture does not boil or over-thicken.
Remove the hot pudding from the heat, and stir in the remaining one teaspoon of vanilla.
Pour the pudding through a fine mesh sieve into a bowl, to remove any clumps from the pudding.
Cool slightly, then cover by pressing a sheet of plastic wrap directly on the surface of the pudding. This is optional, but it helps to prevent a skin from forming on the pudding.
Chill the pudding in the fridge for at least 3-4 hours, or until it's firmed up slightly.
To serve, top the banana pudding with sliced bananas, coconut whipped cream, and toasted coconut flakes.