1. Preheat oven to 425°F. Place banana (in its peel) on a baking sheet, and bake in preheated oven until peel is very dark and just starting to split open, 20 to 25 minutes. Let banana cool until it deflates slightly and is cool enough to handle, about 10 minutes.
2. Peel banana; transfer banana flesh to a food processor, and process until smooth.
3. Process ice cream, milk, 2 tablespoons of the banana puree, and peanut butter in a blender until smooth. Pour mixture into a tall glass; add a dollop of peanut butter, and serve with a straw.