Thoroughly grease and flour a large bundt pan. Preheat your oven to 350 degrees.

In a large stand mixer or mixing bowl, beat the eggs for about 30 seconds. Add the oil and sugar and mix well.

Add the buttermilk, lemon juice, and lemon zest and mix well.

Add the flour, salt, and baking powder all at once. Beat on low speed just until all the flour is incorporated and most of the lumps are gone.

Pour the batter into your prepared bundt pan and bake for 40-45 minutes until the top is golden brown and a knife or toothpick comes out clean when inserted into the center.

Allow to cool for 30 minutes in the pan. Then loosen the sides and turn the cake out onto a wire rack set over a plate to cool completely.

Add the 1/2 cup sugar and 1/2 cup lemon juice to a small saucepan on the stove. Cook over medium heat, stirring constantly, until the sugar dissolves. Remove from the heat.

Spoon or brush all over the top and sides of the warm cake. Use all the syrup. Allow the cake to cool completely at room temperature.

In a stand mixer, or in a large bowl with a hand mixer, whip the softened cream cheese and butter on high speed until smooth and creamy.

Add the powdered sugar, lemon zest, milk, and pinch of salt and blend on low speed until the powdered sugar disappears. Then increase the speed and beat for about two minutes. Add more powdered sugar or milk if necessary to reach your desired consistency.

Pipe over the top of the cake and decorate as desired. Store the cake in the refrigerator until ready to serve.