Preheat an oven (with one of its racks positioned in the center) to 325°F. Generously coat the inside of a 10-inch bundt or tube pan or two 8-inch loaf pans with butter. Dust the inside of the pan lightly with flour, shaking and tapping it gently to remove any excess.
In a small bowl, whisk together the flour, salt, baking powder, and baking soda and set it aside. In a liquid measuring cup, stir together ¼ cup of the lemon juice, the buttermilk, and vanilla, and set it aside.
In a stand mixer fitted with the paddle attachment, cream the butter on medium speed until smooth, about 1 minute. With the mixer running on medium speed, gradually add 2 cups of the sugar, 1 cup at a time, until incorporated. Continue beating until light and fluffy, 3–5 minutes more. Using a silicone spatula, scrape down the sides and bottom of the bowl, then, with the mixer back on medium speed, add the eggs, one at a time, beating well after each addition, until the eggs are completely incorporated. Add the lemon zest, and then, with the mixer on low speed, beat in the flour mixture in three additions, alternating with the lemon juice-buttermilk mixture (beginning and ending with the dry ingredients). Once again, scrape down the sides and bottom of the bowl, then continue mixing just until the batter is smooth. Pour the batter into the prepared pan, transfer to the oven, and bake until a wooden skewer inserted in the center comes out clean, 60–70 minutes. Remove from the oven and cool slightly in the pan.
Meanwhile, make the lemon syrup: To a small pot, add the remaining ½ cup sugar and ½ cup lemon juice. Bring to a boil, then lower the heat to simmer and cook until the sugar is dissolved and the syrup is thickened slightly, about 1 minute. Remove from the heat and stir the syrup for a few minutes to cool slightly.
Place a wire cooling rack over a large rimmed baking sheet and set it aside. Use a wooden skewer to poke holes all over the bottom of the cake, then gradually spoon the lemon syrup over the entire surface, allowing the cake to absorb the syrup between spoonfuls. Repeat until all of the syrup is used. Let the cake cool in the pan for 10 minutes, then turn it out onto the wire rack and set aside to cool completely.
Meanwhile, make the lemon glaze: In a liquid measuring cup or medium bowl, whisk together the lemon juice and confectioners’ sugar until smooth. When the cake is cooled, drizzle the glaze over the top of the cake, allowing it to drip down the sides. Set aside until the icing is dried before slicing and serving.