Slowly heat milk in a sauce pan on medium low heat. Do not let it boil, just get it nice and steamy. Meanwhile, whisk sugar and eggs together in a bowl. When milk is hot, slowly...and I mean slow as you can...drizzle some of the milk into the sugar/egg mixture, stirring constantly.
Pour the milk, eggs and sugar mixture back into the sauce pan and heat on low until thickened, again stirring constantly. It should reach the consistency of liquid pudding.
Remove from heat and when it reaches room temperature add the cream and vanilla and stir. Refrigerate covered overnight so that the mixture volumizes. Do not short cut this step.
For the bananas:
Preheat oven to 400 degrees. On a foil lined pan place the sliced bananas and sprinkle with sugar, cinnamon and drizzle with the butter. Bake for 20 minutes. Let mixture cool then puree in a food processor.
When completely cooled, add the caramelized bananas to the ice cream mixture and stir to combine. Churn according to your ice cream maker's instructions.
Notes
Tip: Don't wait too long to transfer the caramelized bananas to the food processor. If allowed to cool too much, the brown sugar will start to hardenAs with most home ice cream makers, the final consistency of the ice cream will resemble soft serve. Place in the freezer for a couple of hours to achieve a texture to scoop.