In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl.

In a medium bowl, sift together flour, baking powder, and salt. In a small bowl, combine milk and rum. With mixer on low speed, gradually add flour mixture to butter mixture alternately with milk mixture, beginning and ending with flour mixture, beating just until combined after each addition. Transfer batter to a large bowl; set aside.

Clean bowl of stand mixer. Using the whisk attachment, beat egg whites at high speed until stiff peaks form, about 2 minutes. Gently fold half of egg whites into batter. Fold in remaining egg whites. Pour batter into prepared pan, smoothing top if necessary.

Bake until a wooden pick inserted in center comes out clean, 25 to 30 minutes. Let cool in pan for 15 minutes. (You do not want the cake to cool completely. If completely cool, it will affect the bake time for the meringue.) Run a knife around edges of pan. Open pan, and place strips of parchment paper around cake, letting an excess of 2½ inches extend up sides. Close pan back around cake.

Spoon about one-third of Vanilla Meringue on top of cake. (You want to handle the meringue as little as possible, so this does not need to be a measured amount.) Spread into an even layer. Spoon remaining Vanilla Meringue in center of cake, and gently spread around center of cake so there is less meringue on outside edge, leaving about a ¼ inch border between meringue layer and cake pan. (If there is too much meringue pushed to outside edge, the meringue may collapse when out of the oven.)

Return cake to oven, and loosely cover with foil; immediately reduce oven temperature to 325°F (170°C). Bake for 15 minutes. Reduce oven temperature to 300°F (150°C), and bake until meringue is dry to the touch, 30 to 35 minutes more. Let cool completely in pan. Serve with Brown Sugar Rum Sauce.