For the banana bread: Preheat oven to 350 degrees. Place bananas on a parchment-lined sheet pan and pierce a few times with a fork or skewer. Cook for 10 to 12 minutes, until the banana peels are black. Remove pan from oven. Once they have cooled enough to touch, peel and mash as thoroughly as possible.
In a medium bowl, stir together the flour, baking soda, and salt. In the bowl of a stand mixer, beat the eggs and sugar on medium speed for 5 minutes. In a slow and steady stream, add the melted butter, followed by the oil, into the egg mixture until blended.
Add the buttermilk and vanilla bean paste to the mashed bananas and whisk vigorously. Stir the banana mixture into the egg mixture. Gently fold in the dry ingredients into the batter until combined but not over mixed. Fold in the nuts, if using.
Pour the batter into a greased loaf pan and cook for 1 hour. Let cool on wire rack before removing from pan.
For the sandwiches: Preheat a gas or charcoal grill (or an indoor grill pan) on medium-high heat. Cut off the crust of the ends and slice the banana bread crosswise into 16 thin pieces. Place the banana bread slices on the grill and cook until dark grill marks form, 2 to 3 minutes. Flip and cook for an additional 2 to 3 minutes. Allow bread to cool before making the sandwiches.
Remove the ice cream from the freezer and thaw until spreadable but not melted, 5 to 10 minutes. Scoop a generous amount of ice cream (about 1 cup) on a slice of grilled banana bread and spread evenly across the surface to make 1/2-inch of ice cream filling. Top with another slice of banana bread, pressing firmly to even out the ice cream. Wipe excess ice cream from the edges and wrap sandwich tightly with plastic wrap. Freeze sandwich and continue with remaining sandwiches. (You may need to put the ice cream carton back in the freezer a few times throughout the process to keep it from getting too runny.)
Chill the sandwiches until the banana bread is frozen, at least 1 hour. Remove sandwiches from plastic wrap. Using a serrated knife, cut off a thin piece from all four sides to create a smooth, clean-looking sandwich. Allow to thaw for a few minutes before serving, to soften the bread and ice cream.
The banana bread and ice cream sandwiches can be made 2 to 3 days before serving.