Pour milk into a saucepan. Add 2/3 cup plus 1 tablespoon sugar and egg yolks. Whisk thoroughly. Cook over medium heat, stirring gently with a spatula, until custard is steaming-hot and thick enough to coat the back of a spoon, about 10 minutes. Stir in heavy cream and salt and remove from heat.
Pour custard through a fine mesh strainer and refrigerate until completely chilled, 8 hours to overnight.
Pour mixture into an ice cream maker and churn according to manufacturer's instructions, about 20 minutes. Seal top with plastic wrap, cover with a lid, and freeze until firm, at least 4 hours.