Whisk sugar, flour, and salt in a medium saucepan until combined.
Add egg yolks and whole egg, whisk until smooth.
Slowly whisk in milk and heavy cream until incorporated.
Place over medium heat and whisk constantly for 8-10 minutes until thickened and pudding coats the back of a spoon.
Remove from heat and whisk in vanilla extract.
Pour through a fine mesh sieve into a large bowl.
Press plastic wrap directly onto pudding surface.
Refrigerate for at least 2 hours until completely set.