In a medium mixing bowl, using a hand mixer, beat together the sugar and eggs until thickened and pale yellow, about 3 minutes. Add the flour and salt and beat until combined. Set aside.
In a medium saucepan, heat the milk just to boiling.
With the hand mixer on a medium speed, add the milk to the egg mixture in a slow steady stream. Mix until well combined. Transfer the egg mixture to the pan and heat over medium heat, stirring constantly until mixture thickens. Remove from heat and pour custard through a fine sieve into a large clean bowl to remove any egg solids.
Add the bananas to a mini prep food processor and pulse until pureed.
Stir the banana puree, cream and vanilla into the custard. Cover with plastic wrap and refrigerate until well chilled, 2 hours or overnight.
Heat oven to 325. Spread walnuts into a single layer on a baking sheet. Bake for 8-10 minutes until fragrant and lightly toasted.
Stir the chilled custard, then churn in your ice cream machine, according to the manufacturers instructions. Halfway through churning pour in the walnuts and chocolate chunks. Continue to churn until ice cream is ready.
Ice cream will be soft at this point, but ready to eat. Otherwise, transfer the ice cream to a freezer-safe container and chill until firm, at least two hours.