Preheat the oven to 325°F.
Begin by caramelizing a 1-quart souffle dish or a 6-inch-diameter cake pan; In a small heavy-bottomed saucepan, combine 1/2 cup of the sugar, and 3 tablespoons of water. Cook the mixture over medium-high heat until the sugar caramelizes-it will have a rich golden-brown hue. Immediately remove the saucepan from the heat. Working quickly, pour the caramel into the souffle dish or cake pan. Using potholders to protect your hands, tilt the dish in all directions to coat the bottom and about one inch of the adjacent sides. Continue tilting the dish until the caramel has hardened, then set the dish aside.
To prepare the custard, put the milk into a heavy-bottomed saucepan over medium heat. As soon as the milk reaches a boil, remove the pan from the heat and set it aside. In a bowl, whisk together the eggs and the remaining 1/3 cup sugar, then stir in the rum, vanilla, cardamom or cinnamon, banana puree, and the remaining 2 teaspoons of lemon juice. Stirring constantly to avoid curdling the eggs, pour the hot milk into the banana mixture. Transfer the custard to the caramelized dish.
Set the custard dish in a small roasting pan and pour enough hot tap water into the pan to come 1 inch up the sides of the custard dish. Bake the flan until a knife inserted in the center comes out clean-20 to 30 minutes. (Take care not to insert the knife so deep that it pierces the caramel coating.) Remove the flan from the hot-water bath and let it cool to room temperature. Put the flan into the refrigerator until it is chilled- about 3 hours.
To unmold the chilled flan, invert a serving plate over the top of the dish, then turn both over together. The dish should lift away easily; if it does not, turn the dish right side up again and run a small sharp knife around the top of the custard to loosen it. If you like garnish the flan with a ring of banana slices. Cut the flan into wedges and spoon some of the caramel sauce over each one before serving.