In a medium bowl, combine bananas and milk; cover and refrigerate for at least 8 hours or up to overnight.

Strain milk into a medium saucepan, discarding solids. Heat milk over medium heat until steaming. (Do not boil.)

In another medium bowl, whisk together sugar, cornstarch, and salt; whisk in egg yolks until smooth. Gradually whisk in half of hot milk. Whisk egg mixture into remaining hot milk in pan. Bring to a boil over medium heat, whisking frequently. Cook, whisking constantly, until thickened and an instant-read thermometer registers 185°F (85°C), about 3 minutes. Remove from heat.

Whisk in 2 tablespoons (28 grams) butter in two additions until melted and smooth; whisk in vanilla bean paste.

Strain mixture through a fine-mesh sieve into a large bowl, discarding solids. Cover with plastic wrap, pressing wrap directly onto surface of pastry cream to prevent a skin from forming. Refrigerate until thick and cold, at least 4 hours, or overnight.

Let pastry cream and remaining ½ cup (113 grams) butter stand at room temperature for 20 to 30 minutes. (Butter should be slightly soft and the same temperature as pastry cream.)

In the bowl of a stand mixer fitted with the paddle attachment, beat softened butter at medium speed until smooth, about 1 minute. Gradually add pastry cream, beating until combined after each addition and stopping to scrape bottom and sides of bowl. Refrigerate until thick but pipable, about 1 hour, stirring every 30 minutes. Use immediately.