In a medium bowl, whisk the egg yolks and honey or sugar 2 to 3 minutes until lighter in color. Set aside.
In a heavy bottomed pot add the milk and cream and heat over medium, whisking often until the mixture begins to simmer. Turn off the heat.
While constantly and rapidly whisking the eggs, add 1 cup of the milk mixture to the eggs in a slow steady stream. You can use a ladle for this if it's easier. Add one more cup of milk while whisking then transfer the egg mixture back to the main pot.
Return to medium/medium low heat and whisk constantly until the mixture thickens and reaches about 175 degrees. It should not boil or even simmer, but should thicken enough that you can draw a line through it on the back of a spoon and it doesn't run back together.
Cool 30 minutes then add the vanilla.
Allow to cool until no longer steaming and transfer to a bowl. Place plastic wrap directly against the surface and refrigerate for 8 hours minimum, but 24 hours is best, but it can stay for up to 48 hours.
When ready to churn, first peel and mash the bananas; crush the cookies and set aside.
Follow the instructions for your ice cream maker; Churn 25 minutes and no longer than 40 to avoid over churning. During the last 10 minutes, add the bananas and cookies and churn to mix in.
Transfer to an air tight container and freeze 3 to 4 hours until fully firm.