Unfold The Sheets Of Puff Pastry.

Place Each One Onto A Baking Sheet.

Press Down On Any Cracks To Repair Them, And Gently Roll The Pastry To Smooth It Out.

Place The Baking Sheets In The Refrigerator For 1 Hour.

Meanwhile, For The Filling, Place The Butter And Sugar In A Large Mixing Bowl, And Beat With An Electric Mixer On Medium Speed Until Light And Creamy, 30 Seconds.

Add The Egg, And Beat Until Creamy, 30 To 45 Seconds More.

Set Aside.

Place The Almonds And Confectioners’ Sugar In The Bowl Of A Food Processor Fitted With A Steel Blade.

Pulse Until The Mixture Resembles Meal.

Turn The Almond Meal Into The Bowl With The Butter And Sugar.

Add The Flour, Rum, And Vanilla, And Stir With A Wooden Spoon Or A Rubber Spatula Until Just Incorporated.

Set Aside.

(If Your Kitchen Is Warm, Cover And Place In The Refrigerator Until Ready To Assemble.)

Remove The Baking Sheets From The Refrigerator.

Paint A 1" Border Of The Egg Wash Around The Edges Of 1 Pastry Sheet.

Spoon The Almond Mixture Into The Center Of The Pastry, Pressing It With The Spoon Just Until It Meets The Egg Wash.

Do Not Let The Filling Get Onto The Egg Wash Border.

Place The Second Pastry Sheet Over The Filling, And Press Down On The Edges Firmly So They Seal.

Brush The Top With Egg Wash, And Let The Cake Rest For 1 Hour Before Baking.

Place A Rack In The Center Of The Oven, And Preheat The Oven To 400°F.

Bake Until The Cake Is Puffed And Brown, 40 To 45 Minutes.

Slice And Serve.