Separate egg yolks into a bowl and set aside.
In a saucepan set over medium heat, combine milk, heavy cream, and bananas. Continue to cook stirring frequently until bubbles just begin to form along the edges of the pot. Do not let the milk boil.
While the milk mixture is heating, in a large bowl, whisk together the egg yolks and sugar.
Slowly pour one cup of the hot milk into the egg yolk mixture, whisking constantly. It may be helpful to set the bowl onto a kitchen towel to stabilize it while you whisk. Pour this mixture back into the saucepan and continue to stir over low heat until the temperature reaches 185 degrees F. The mixture should be slightly thickened, enough to coat the back of a spoon. Do not allow the mixture to come to a boil.
Remove the milk from the heat and stir in vanilla bean paste.
Pour the custard through a fine mesh strainer into a clean bowl set over an ice bath and allow it to cool completely; stirring frequently to help the cooling process along.
Once completely cooled, pour the custard into the bowl of your ice cream machine and churn for 20 minutes or until set. Be sure to refer to the directions of you specific ice cream maker. This may take longer if your custard was not completely cooled.
Transfer the frozen custard to an airtight container and freeze 3 hours or overnight.