In a medium saucepan, bring the milk, cream, and salt to a simmer over medium heat.
While the liquid is heating, in a medium mixing bowl, whisk the egg yolks, sugar, and cornstarch until well combined. Once the milk mixture is heated, use a ladle and slowly add 1 cup of the liquid to the egg mixture, whisking constantly, to temper the eggs.
Return the tempered mixture back to the saucepan and cook over medium heat until it is thick and bubbly – about 2-3 minutes.
Remove the saucepan from the heat and add in the chocolate chips and vanilla paste (or extract). Mix with the whisk or a rubber spatula until the chocolate has melted, making sure to scrape the bottom and corners of the saucepan to fully incorporate the mixture. Once the chocolate has melted, stir in the butter and mix until melted.
Strain the mousse through a fine mesh strainer set over a medium bowl – scraping the bottom of the strainer to make sure you have all of the custard! Transfer the custard to the cooled pie shell and spread evenly into all corners with a spatula. Cover with plastic wrap and let the chocolate pie cool in the refrigerator for at least 1 hour, for the custard to set.