Preheat the oven to 350 degrees F. Grease a 9-inch square cake pan with 1 tablespoon of the butter.
For the cake: In the bowl of an electric mixer fitted with a whisk attachment (or an electric hand mixer, or by hand), cream the 1 stick of butter and granulated sugar together until smooth, 5 to 8 minutes. As you are creaming the butter and sugar, scrape the sides down with a rubber spatula so you are sure to be combining the butter and sugar as it mixes.
In a separate bowl, sift together the flour, baking soda, and salt. Set aside.
When the butter and sugar are integrated, blend in the yogurt. Add the eggs, one by one. Add the lemon zest, lemon juice, and vanilla. Remove the bowl from the mixer and fold in the dry ingredients. Transfer the batter to the prepared pan and bake, undisturbed, until a toothpick inserted into the center of the cake comes out clean, 40 to 45 minutes. Allow the cake to cool for 10 minutes before inverting it on to a plate and allowing it to cool for an additional 20 minutes.
For the frosting: In a small saucepan, combine the water and granulated sugar and bring to a simmer. Stir to make sure all of the sugar has dissolved. Remove from the heat and transfer to a bowl to cool. Stir in the lemon zest, lemon juice, nutmeg, and cinnamon. Stir to blend. Set the syrup aside to cool.
In a small saucepan, heat the cream. Place the chocolate chips in a medium bowl. When the cream simmers, pour it over the chocolate chips. Whisk to blend. Set aside to cool.
In the bowl of the food processor, combine the confectioners' sugar, butter, and almond extract. Pulse to blend. Continue blending for about 30 seconds until the ingredients are mixed together. Use a rubber spatula to scrape every bit of the chocolate mixture into the processor and mix again to blend. Transfer the frosting to a bowl and allow it to sit a little before frosting the cake.
Transfer cake to a serving platter. Use a pastry brush (or spoon) to put some of the syrup on the cake. Do not make too wet or the cake will be soggy and fall apart, use just a little bit here and there. Carefully spread the frosting over the cake starting in the center and working it to and around the edges. Make sure there is enough frosting to coat the top and sides of the cake. Serve.