Preheat an oven to 300 degrees F (150 degrees C). Grease and flour an 8 inch round baking pan.
Beat the butter and 1 1/2 cups of sugar with an electric mixer in a large bowl until light and fluffy. The mixture should be noticeably lighter in color. Add the room-temperature egg yolks one at a time, allowing each egg yolk to blend into the butter mixture before adding the next. Pour in the flour alternately with the milk, mixing until just incorporated. Stir in the almond extract. Beat egg whites until stiff peaks form in a large glass or metal mixing bowl. Lift your beater or whisk straight up: the egg whites should form a sharp peak that holds its shape. Fold the egg whites into the batter; mixing just enough to evenly combine. Pour the batter into prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 90 minutes. Cool in the pan for 10 minutes before removing to cool completely on a wire rack. Cut any excess cake so that it is flush with the sides of the pan.
Bring the water and 1 cup of sugar to a boil for 8 to 10 minutes. Reduce heat to a simmer, then add brandy and 1 teaspoon vanilla extract. Simmer for 2 to 3 minutes and remove from heat. Allow syrup to cool to room temperature. Prick holes in the cooled cake with a toothpick, then slowly saturate the cake with the brandy syrup.
