Whisk eggs together with sugar until fluffy and pale yellow. Add cornstarch and salt, whisk until incorporated and smooth. Set aside.
In a small saucepan, bring milk and cream just to the start of a boil. Remove from heat, add tea bags, steep for 15 minutes. Remove tea bags, squeezing out excess liquid back into the pot. Bring cream up JUST to a boil once again.
Measure about ¼ cup of the hot cream mixture, and stream slowly into egg mixture while whisking. Continue streaming liquid and whisking until it is completely incorporated, and mixture is smooth. Repeat with another ¼ cup of hot cream.
Remove saucepan from heat, pour remaining egg mixture into cream mixture, whisking constantly. Once fully incorporated and smooth, return to heat. Turn heat to medium-low. Continue whisking mixture constantly, cooking until mixture is very thick. Remove from heat, stir in butter until melted and smooth. Cool to room temperature.
Spoon filling into prepared crust, cover with plastic wrap. Chill until set, about 3 hours. To serve, top with whipped cream.