Line a medium shallow baking dish or bowl with plastic wrap; place a fine-mesh sieve inside.

In a medium saucepan, heat milk over medium heat until steaming. (Do not boil.)

In a medium bowl, whisk together sugar, cornstarch, flour, and salt. Whisk in egg yolks until well combined. (Mixture will be very thick.) Gradually add half of warm milk to sugar mixture, whisking constantly. Add sugar mixture to remaining warm milk in pan. Bring to a boil over medium heat, whisking constantly. Cook, whisking constantly, until thickened and mixture no longer tastes starchy, 3 to 4 minutes. Remove from heat.

Add butter to warm milk mixture in two additions, stirring until combined; stir in vanilla bean paste. Pour and press mixture through fine-mesh sieve into prepared dish or bowl; smooth into a thin layer. Cover with a piece of plastic wrap, pressing wrap directly onto surface of pudding to prevent a skin from forming. Refrigerate until thoroughly chilled, 2 to 3 hours or up to overnight. Stir until softened and smooth before using.