In a medium-sized mixing bowl, combine flour, sugar, and salt.

Introduce the cold butter cubes to the flour mixture. Utilize a pastry cutter to integrate the butter until a crumbly texture is achieved.

Gradually add cold water to the mixture, starting with 1/4 cup, and adding more if required, until dough formation.

Shape the dough into a ball, encase with plastic wrap, and refrigerate for approximately 50 minutes.

In a separate bowl, blend ½ cup of brown sugar with the flour. Incorporate the egg, egg yolks, and milk into the mixture, ensuring a smooth consistency.

In a heavy pan over medium heat, dissolve the remaining brown sugar until it acquires a golden hue.

Carefully add 1 tablespoon of hot water to the dissolved sugar, consistently whisking. Gradually blend in the egg-milk mixture and continue to cook until the filling thickens.

Once removed from the heat source, incorporate vanilla extract and butter.

Preheat the oven to 425°F (220°C).

On a floured surface, roll out the refrigerated dough to a thickness of approximately 1/4 to 1/3 inch. Position in a pie plate.

Line the formed crust with parchment paper, introduce pie weights, and bake for approximately 15-20 minutes.

Upon completion, reduce the oven temperature to 350°F (175°C).

Pour the prepared filling into the pre-baked crust and return to the oven for an additional 15-20 minutes. Allow the pie to cool upon removal.

Simultaneously heat a medium pot filled with 2 inches of water and position a medium bowl atop. To the bowl, add the listed ingredients for the meringue.

Whisk the contents consistently, ensuring the mixture reaches a temperature of 160°F (71°C).

Transfer the heated mixture to a mixer and whisk on medium-high speed until medium peaks form.

Gently apply the prepared meringue atop the cooled pie. Optionally, utilize a kitchen torch to toast the meringue to preference.