Step 1: Preheat oven to 350 degrees F. Grease a 9-inch tart pan, or line with parchment paper.
Step 2: Make the crust: Whisk together the flour, powdered sugar, and salt. Add softened butter and vanilla. Mix together until a ball forms. (A food processor works great for this.)
Step 3: Press dough evenly in bottom of tart pan, and all the way up the sides. Prick with a fork a few times around the bottom of crust. Bake for 15 minutes, or until just starting to brown. Remove from oven and set aside to cool for a few minutes.
Step 4: Make the filling: In a medium saucepan whisk together eggs, egg yolks, sugar, and salt. Stir in lemon juice and zest. Place saucepan over medium heat and cook, stirring constantly, until a thick curd forms, about 10-12 minutes. It should be the consistency of pudding.
Step 5: Remove pan from heat and stir in the butter and heavy cream.
Step 6: Strain lemon filling through a fine-mesh strainer, then pour into the crust.
Step 7: Bake tart until set, but still slightly jiggly in middle, about 15 to 18 minutes.
Step 8: Cool on a wire rack for one hour, then chill in fridge for at least 3 hours.
Step 9: Sprinkle powdered sugar over top before serving, if desired.