Add hot milk to egg mixture in a slow stream, whisking constantly, then pour custard back into saucepan. Cook over medium-low heat, stirring constantly, until thickened and registers 170°F (do not let boil). Immediately strain through a fine-mesh sieve into a clean bowl and stir in butter and vanilla. Chill custard, its surface covered with parchment paper (to prevent a skin from forming), until cold, 2 to 3 hours.