Set an oven rack to the lowest position and preheat your oven to 350°F. Place the figs, half-and-half, and rum in a medium saucepan and bring to a boil. Turn off the heat, cover the pan, and steep the figs for 10 minutes.

Place the condensed milk, eggs, cream cheese, and fig jam in a blender and blend until smooth, about 2 minutes. Add the half-and-half, figs, and rum, and blend until they are completely incorporated.

Bring the sugar, water, and cinnamon stick to a simmer in a medium, heavy-bottomed saucepan and remove the pan from the heat. Stir until the sugar is dissolved and the syrup is dear. Return the saucepan to the heat and bring the liquid to a boil, swirling the pan every now and then, until the syrup caramelizes to a deep brown, 4 to 5 more minutes. Immediately pour the caramel into a 2½ quart metal charlotte mold or a 9 × 5-inch loaf pan. Carefully remove the cinnamon stick with a spoon or tongs. Tip the mold or pan to coat the bottom and sides with the caramel. Let it cool for a few minutes and then pour the custard mixture into it.

Line an 8-inch square baking pan (or a larger rectangular baking pan, if using a loaf pan) with a doubled kitchen towel Put the mold in the baking pan on top of the towel and then place the pan into the oven. Use a cup to add hot water to the baking pan, adding enough water to reach the middle of the mold or loaf pan. Bake the flan until the custard is set but still jiggles when shaken and a skewer stuck into the flan about 1 inch from the pan edge comes out dean, about 1 hour and 25 minutes.

Carefully lift the mold out of the pan. Turn off the oven and let the water in the baking dish cool a little before removing it. Refrigerate the flan until it is completely chilled.

To serve, set the mold over direct heat until the bottom gets hot, about 1 minute. (This is to melt the bottom layer of caramel so that the flan will slip out of the mold.) Run a knife around the edge of the flan to loosen it from the mold. Place a serving platter over the mold and then invert the mold onto the platter. Lift off the mold. Cut the flan into wedges or slices and serve.