Bring 1 cup of the milk to a boil in a medium saucepan over medium heat.
Meanwhile, combine the sugar, cornstarch and salt in a medium bowl. Whisk in the remaining 1 cup milk a little at a time to avoid lumps, then whisk in the egg.
When the milk is boiling, remove the saucepan from the heat and gradually whisk in the egg mixture, stirring constantly. Return to medium-low heat and cook, whisking constantly, until the pudding comes to a simmer and is thick and creamy, 5 to 7 minutes. Remove from the heat and whisk in the butter and vanilla until smooth. Stir in the chocolate chips until smooth.
Divide the pudding among four to six 4-ounce bowls or 1/2-cup ramekins, topping each with 1 to 2 tablespoons of the berries. Cover with plastic wrap and refrigerate until fully set, about 2 hours.
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