Position a rack in the center of the oven and heat the oven to 350°F. Butter eight 6-ounce ceramic ovennproof ramekins or Pyrex custard cups and arrange them in a baking dish or roasting pan (a 10x15-inch or two 8x8-inch Pyrex dishes work well).
In a large bowl, whisk the melted butter with the brown sugar and egg yolks until smooth, about 1 min. Add the flour and salt and pour in just enough milk to whisk the flour smoothly into the egg yolk mixture. Then whisk in the remaining milk, along with the rum and vanilla, until smooth. The mixture will be very fluid.
Put the egg whites in a large bowl. Beat with an electric mixer (a hand-held or a stand mixer fitted with the whisk attachment) at medium speed until the egg whites begin to foam, 30 to 60 seconds. Increase the speed to high and beat just until the egg whites hold soft peaks when the beater is pulled away from the whites, another 1 to 2 min. Reduce the mixer speed to medium. With the mixer running, very slowly sprinkle in the granulated sugar; this should take about a minute. Stop the mixer and scrape the bowl. Beat on high speed until the whites hold medium-firm peaks when the beater is pulled away, about another 30 seconds.
Scrape one-third of the egg whites onto the egg yolk mixture and whisk until combined. Gently incorporate the remaining egg whites evenly into the batter, using the whisk in a folding/stirring motion. The batter will still be thin.
Portion the mixture evenly among the ramekins; the cakes don’t rise much, so you can fill the ramekins to within 1/8 inch of the top. Pull out the oven rack and put the baking dish full of ramekins on the rack. Pour warm water into the dish to reach halfway up the sides of the ramekins. Bake until the tops of the cakes are light golden and slightly puffed and, when touched with a finger, they should feel spongy and spring back a bit but hold a shallow indentation, 25 to 30 min. Using tongs, carefully transfer the ramekins from the water bath to a rack. Let cool to room temperature and then refrigerate for at least 2 hours and up to 24 hours before serving, with whipped cream if you like.