Preheat the oven to 350 degrees F. Grease a loaf pan with butter. Set that aside.
In a small bowl, mix together the flour, salt, and baking powder. Set that aside.
In the bowl of a stand mixer fitted with the paddle attachment, add the granulated sugar and butter. Cream on medium speed for 1-2 minutes until fully combined and light and fluffy.
Add the eggs, sourdough discard, lemon zest, and lemon juice to the bowl of the stand mixer. Mix well.
Add the dry ingredients to the bowl with the wet ingredients. Mix for 30 seconds or until the ingredients are just mixed. Do not overmix at this point.
Pour the batter into the prepared loaf pan. Bake for 50-55 minutes or until a toothpick is inserted and comes out clean. Remove the loaf pan from the oven and let it cool for 10 minutes.
While the loaf cools, mix together the ingredients for the lemon syrup. Remove the cake from the loaf pan and place it on a cutting board or a plate. Brush the lemon syrup all over the loaf while it is still warm. After the syrup has been brushed into the loaf pan, let it cool to room temperature.
When the loaf has cooled completely, mix together the glaze ingredients. For a thicker glaze, add more powdered sugar. For a thinner glaze, add more milk or lemon juice. Pour the glaze over the cake and serve immediately.