How to Make It
Preheat oven to 350°F. Place a kitchen towel on a work surface and set a 9 1/2-inch deep-dish pie plate on towel. Place sugar in a large, heavy stainless steel skillet. Cook over medium until sugar becomes fragrant and begins to melt around edges of pan, about 6 minutes. Gently swirl pan to incorporate melted and granulated sugar. Continue swirling pan until all of the sugar is completely melted and has darkened to the color of dark brown sugar, about 10 minutes. Immediately pour melted sugar into prepared pie plate, coating the bottom evenly. Let stand until cool, about 20 minutes.
Meanwhile, fill a large, 2-inch-deep roasting pan with hot water to a depth of about 3/4 inch (enough so it will not overflow when flan dish is added), and place pan in preheated oven. Whisk eggs in a large bowl. Add whole milk, sweetened condensed milk, and vanilla bean seeds, and whisk until smooth.
Set pie plate in hot water bath in oven, and carefully add milk mixture. Bake until edges are set and center looks glossy and retains a slight wiggle, about 50 minutes. Remove pie plate from water bath. Cool completely on towel, about 2 hours. Run a knife around edge of flan to loosen. When pan is rotated, flan should float in caramel. Press plastic wrap directly onto surface; chill at least 3 hours or preferably overnight.
Set pie plate in a shallow dish of hot water to soften caramel, about 2 minutes. Invert flan onto a large rimmed platter.
Beat cream and rum with a heavy-duty electric stand mixer fitted with the whisk attachment on medium-high speed until medium peaks form, about 2 minutes. Spoon any remaining caramel sauce from pie plate over flan, and serve with dollops of whipped cream on the side.