Put the split vanilla bean and its seeds in the saucepan, then add the milk and ½ cup (100 g) of the sugar. Cook over medium heat, stirring occasionally, until the thermometer registers 170° F/77° C. Remove from heat, cover, and allow to steep for 30 minutes.
Remove the vanilla bean pod and return the pot to the burner. Warm over medium heat, stirring occasionally to prevent scorching, until the thermometer registers 170° F/77° C.
In the other 3-quart bowl, whisk the egg yolks with the remaining ¼ cup (50 g) sugar until the mixture is foamy and slightly thickened.
Temper the egg yolks by adding about half of the hot milk mixture one ladle at a time, whisking continuously. Whisk the tempered egg mixture into the saucepan and return to the burner. Cook over medium heat, stirring continuously with a wooden spoon or heatproof spatula until the thermometer registers 185° F/85° C. Do not allow the mixture to boil. Remove from heat and insert the immersion blender into the hot liquid. Blend until smooth. (Note: You can also use a whisk and vigorously whip the mixture, or use a blender.)
Make an ice bath in the large bowl and set the bowl with the heavy cream mixture on top. Place a fine-mesh strainer on the rim of the bowl and carefully pour the custard through the strainer. Stir the custard about every 5 minutes until the mixture is cooled, about 30 minutes. Remove the bowl from the ice bath, dry the bottom, and stir in the mashed banana. Cover with plastic wrap and chill in the refrigerator for at least 8 hours or overnight.
About 30 minutes before churning, set the bowl in the freezer to get the mixture as cold as possible without starting to freeze. Pour the mixture into your ice cream maker and follow the manufacturer’s directions for processing. During the final 30 seconds of churning, drop spoonfuls of the Nutella, or you can pipe it in with a sandwich bag if you want smaller bits.
Transfer the churned gelato to the plastic container. Press a layer of plastic wrap against the gelato and seal the lid on top. Freeze until fully hardened, at least 2 hours. To serve, allow the gelato to soften slightly on the counter for about 2 to 3 minutes before scooping.