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To make the syrup: Wash the orange, then zest it into the bowl of a food processor (you’ll use the zest in the batter).
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In a small (1-quart) saucepan (see “tips” below for guidance on substituting a larger pan), bring the water and sugar to a boil over high heat, stirring to dissolve the sugar.
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Carefully lower the zested orange into the saucepan, reduce the heat to medium, and cook, adjusting the heat as needed to maintain a low boil, for 20 to 25 minutes. Keep an eye on the mixture and turn the orange, occasionally at the beginning and more frequently as the liquid starts to evaporate, to ensure all sides are submerged.
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To make the cake: While the orange is boiling, preheat the oven to 350°F with a rack in the center. Lightly grease an 8" round pan that’s at least 2" deep and line the bottom with a parchment round.
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Transfer the boiled orange to a cutting board and let cool slightly. Strain the syrup through a fine-mesh sieve into a heatproof liquid measuring cup or bowl and set aside; you will use some of it for the frosting.
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Cut the orange into large wedges, teasing out any visible seeds, and add to the zest in the food processor. Add the sugar, salt, and butter and process until the butter has melted and the mixture is mostly smooth (some small bits of orange are OK), about 1 to 2 minutes.
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Add the egg and vanilla and pulse 5 to 6 times, until incorporated.
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Add the flour, baking powder, and cardamom and pulse 10 to 12 times, until incorporated.
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Transfer the batter to the prepared pan and smooth the top with a small offset spatula.
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Bake the cake for 34 to 38 minutes, until starting to pull away from the sides of the pan and a toothpick inserted into the center comes out clean.
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Allow the orange cake to cool in the pan on a wire rack for 10 to 15 minutes before turning it out onto the rack to cool completely before frosting.
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To make the frosting: In a medium bowl, mix all the frosting ingredients gently with a spatula until combined, then mix vigorously until smooth.
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Place the cooled cake, bottom side up, on a serving plate. (This gives you a perfectly flat surface for frosting.) Dollop the frosting onto the middle of the cake and use a small offset spatula to gently spread it evenly across the surface.
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Storage information: Store any leftover orange cake, covered, at room temperature for up to 4 days; freeze for longer storage.