Heat 2 1/2 cups half-and-half in a large saucepan over medium heat until small bubbles begin to form around the edges. Remove from heat and set aside.

Beat egg yolks and sugar together in a large bowl until smooth. Slowly stir in warmed half-and-half, stirring constantly. Transfer egg yolk mixture to the same saucepan; heat over medium heat until bubbles begin to form around the edges. Transfer to the same bowl; stir in heavy cream.

Combine bananas and the remaining 3/4 cup of half-and-half in a blender until smooth. Mix the banana puree with the cream mixture, then cover and refrigerate until chilled.

Pour the mixture into an ice cream maker and churn using the ice cream setting. Refrigerate any leftover mixture until freezing. Fold in walnuts, transfer to a 1-quart container, and freeze for 8 hours or overnight before serving.