Preheat oven to 350 F/177 C. Grease and flour a Bundt pan or two bread loaf pans.
Cream the butter and sugar in your mixer for about 5 minutes, until light and fluffy. Add the lemon zest and eggs, one at a time, mixing well in between each addition.
Combine lemon juice, buttermilk (or sour milk), and vanilla. Add one cup of the flour, plus the salt, baking powder and baking soda, to the bowl of the mixer, then add half of the milk mixture. Mix in another cup of flour, and then the rest of the liquid ingredients. Then mix in the last cup of flour.
Pour batter into the prepared pan(s) and smooth the top. Bake at 350 F/177 C for 45-60 minutes, until a toothpick inserted into the center comes out clean.
While the cake is baking, make the lemon syrup. Combine the sugar and lemon juice in a small pan and heat over low heat until the sugar is completely dissolved. Set aside until the cake is done.
Allow cake to cool for about 10 minutes, then use a toothpick or small skewer to poke holes in the top of the cake (if you're using a Bundt pan, you'll be poking holes in the bottom of the cake, since that's the end that's facing up). Then spoon or brush the syrup over the holes while the cake is still in the pan. Place a plate on the top of the Bundt pan, and flip the cake pan and plate over to remove the cake from the pan. Brush any leftover syrup over the top of the cake, then let it cool completely. For bread loaves, brush the syrup equally over the tops of the two loaves while they're still in the pans. Then turn them out of their pans and allow them to cool before glazing.
Once the cake is cool, mix the confectioners' sugar and lemon juice in a small bowl. Drizzle over the top of the cake or breads, allowing it to drip down the sides.