Preheat the oven to 350° F. Butter a 7- to 8-inch 2-inch-deep round cake pan with 1 teaspoon of the butter and set aside.

Combine the remaining 2 tablespoons butter, the rice, milk, nutmeg, cinnamon, vanilla, lemon zest, and 1/2 cup of the sugar in a 2-quart pot, place over medium heat, and cook for 15 minutes. (If the contents of the pot come to a full boil, lower the heat to keep the milk at a low simmer.) Remove from the heat and let the mixture cool for 10 minutes. Beat in the eggs and set the mixture aside.

Combine the remaining 1/4 cup sugar and the water in a small saucepan, place over medium-high heat, and cook until the sugar melts and turns a dark caramel color. Immediately pour the caramel into the center of the prepared cake pan. Do not worry if it doesn't cover the bottom of the pan, as it will dissolve and spread during baking.

Pour the rice mixture into the cake pan and give it a few gentle shakes to distribute the rice. evenly. Carefully transfer the cake pan to the top rack of the oven and bake for 35 to 40 minutes, or until the surface shows some golden areas and the center is just set. Remove the cake pan from the oven and let cool to room temperature.

To unmold, place a plate over the cake pan and invert the plate and pan. Give a few gentle taps to the bottom of the cake pan and carefully remove it from the cake. Place another plate over the cake and invert the plates to turn the cake right side up again. Serve the cake warm or at room temperature and accompany with crème fraîche.