Pour 2 cups milk into a small saucepan and warm over medium-high heat until nearly boiling. Remove from heat, add tea bags, cover and set aside for 10 minutes.
Meanwhile, whisk together sugar, cornstarch and salt in a large bowl. Slowly pour in remaining 1 cup milk and vanilla, stirring to blend.
Remove and discard tea bags. Pour warm milk into bowl with sugar mixture. Stir well and pour back into saucepan. Stir constantly over medium heat until thickened to consistency of mayonnaise. Remove from heat, add chocolate and stir until smooth. Strain into a pitcher or large measuring cup. (You should have about 3 cups.) Pour into ramekins, custard cups or glasses and serve warm, or cover with plastic wrap (let it touch surface of pudding) and refrigerate until ready to serve. Add a dollop of whipped cream, if desired.