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 eggnog and milkjust to the start of a boil.
Measure about ¼ cup of the eggnog 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 eggnog 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, whisk in butter and rum extract until butter is melted and the mixture is smooth.
Cool to room temperature.
Whisk filling again, pour into the cooled crust, cover with plastic wrap. Chill for 2 hours.