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Preheat the oven to 350°F/175°C (gas mark 4). Cut out a round piece of wax/greaseproof paper or parchment paper to line the bottom of a 12-in/30.5-cm round springform or conventional baking pan/tin. Butter the surface of the paper and the walls of the pan. With a pestle, pound the mastic with a pinch of the sugar in a mortar. Set aside.
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In the bowl of an electric mixer outfitted with a whisk,whisk the egg whites at medium speed until foamy. Add 1 1/2 cups/300 g of the sugar and the lemon juice, increase the speed to medium-high, and whisk the whites until they form a stiff meringue. Remove from the mixer and set aside.
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In a clean, large mixing bowl, use the electric mixer with the paddle attachment to whip together the butter and remaining 1/2 cup/100 g of sugar at medium-high speed until light and fluffy, for about 5 to 7 minutes. Add the egg yolks, one at a time, whisking after each addition. Whisk in the milk, brandy, and orange juice. Stir in the orange zest, almonds, pounded mastic, and chocolate.
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Using a rubber spatula, fold the meringue and flour into the egg-yolk-and-liquid mixture, a little at a time, alternating between them.
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Pour the batter into the prepared baking pan/tin. Bake for about 1 hour, or until a thin knife inserted into the center of the cake comes out clean. Remove the cake from the oven and invert onto a wire rack to cool. If making the cake for New Year’s, wrap a small coin in aluminum foil and insert it into the bottom of the cooled cake.
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Turn the cake right-side up and sift confectioners’ sugar over the surface. If the cake is for New Year’s, write the new number with chocolate sprinkles or almonds.