In a large saucepan, whisk together milk, maple sugar, and maple extract. Heat over medium heat until steaming. (Do not boil.)

In a large bowl, whisk together egg yolks, brown sugar, cornstarch, and salt. Gradually add warm milk mixture, whisking constantly. Return mixture to saucepan, and cook over medium heat, whisking constantly, until thickened and boiling, 4 to 5 minutes. Strain mixture through a fine-mesh sieve into a large bowl. Stir in softened butter in two additions. Cover with a piece of plastic wrap, pressing wrap directly onto surface of pastry cream to prevent a skin from forming. Refrigerate until an instant-read thermometer registers 65°F (18°C) to 70°F (21°C), 2½ to 3 hours.

In the bowl of a stand mixer fitted with the paddle attachment, beat room temperature butter at medium speed until smooth, about 1 minute.

Whisk pastry cream until smooth. (At this point, room temperature butter and pastry cream should be same temperature.) With mixer on low speed, slowly add pastry cream to butter, beating until combined after each addition (see Notes) and stopping to scrape sides of bowl. Use immediately.