Heat heavy cream, sugar, and milk in a 2-quart saucepan over medium heat until sugar dissolves, stirring occasionally, about 5 minutes. Remove from heat.

Stir chopped chocolate into warm cream mixture using a wire whisk until smooth and chocolate has melted completely. Whisk eggs and vanilla extract into mixture gradually until well combined.

Place French bread into the bottom of a shallow 8-inch square glass or ceramic baking dish. Pour chocolate mixture over bread. Cover and refrigerate for 30 minutes, stirring occasionally.

Preheat the oven to 300 degrees F (150 degrees C).

Remove pudding from the refrigerator, uncover, and sprinkle with chocolate chips.

Bake in the preheated oven, uncovered, until set, 40 to 45 minutes.

Remove from the oven. Let cool 10 minutes before serving warm, or cover and refrigerate to serve cold later.