To make the crêpe batter, in a bowl, whisk together the eggs and milk until fully incorporated. In a separate bowl, whisk together the flour and the cinnamon. Next, slowly whisk the wet ingredients into the dry ingredients, moving from the inside to outside for a smooth batter. Strain the batter through a fine sieve into a bowl and then whisk in the melted butter and rum. Chill the batter, covered, in the refrigerator for at least 1 hour, and up to 24 hours.
To prepare the bananas, peel them, trim off the stem ends, and halve the fruit lengthwise. Slice each banana half crosswise into three pieces (you’ll have eighteen pieces total); set aside.
Preheat the broiler.
While the broiler is heating, in a stand mixer fitted with the whisk attachment, whip the cream on medium-high speed until it reaches soft peaks. Add 2 Tbs. of the brown sugar and whip until it is thoroughly incorporated. Set aside.
Place an 8-inch nonstick ovenproof skillet over medium heat. Once the pan is warm, add one piece of the cold butter and three banana slices. Sauté the bananas until they are lightly golden on one side, then ladle 1/2 cup of crêpe batter into the pan. Using a rubber spatula, lightly push in the edges of the batter and swirl the pan as though you were making an omelet. Once the crêpe has formed in the pan, flip it over, spread 2 Tbs. of the remaining brown sugar evenly on top of the crêpe, and place the crêpe underneath the broiler. Once all the sugar has been caramelized and is a dark shiny amber color, carefully remove the crêpe from the pan and turn out onto a plate. Repeat the process for the rest of the crêpe batter and bananas.
Serve the crêpes dusted with confectioners’ sugar and topped with whipped cream.