Combine milk, sugar, cornstarch, and salt in a saucepan over medium-high heat. Stir constantly until mixture thickens to the consistency of pudding, 8 to 10 minutes. Remove from heat.

Whisk egg yolks in a bowl until smooth. Add lemon juice and whisk until smooth.

Return pudding to the heat; slowly stir in egg yolk mixture. Increase heat to high and boil, stirring constantly, for at least 2 minutes. Stir in butter and 1 teaspoon vanilla extract. Pour thickened custard into the pie shell.

Preheat the oven to 400 degrees F (200 degrees C).

Combine egg whites, remaining vanilla extract, and brown sugar in a large bowl. Whip with an electric mixture until a fluffy meringue forms. Spread meringue over the custard.

Bake pie in the preheated oven until meringue is golden brown, about 10 minutes. Let cool. Chill for at least 1 hour before serving.