Preheat oven to 350 degrees F.
Beat the butter and sugar in the bowl of a stand mixer or using a hand mixer for 2-3 minutes until light and fluffy. Add the lemon zest and eggs, one at a time. Be sure to scrape the bottom and sides of the bowl.
Whisk the lemon juice, buttermilk, and vanilla together in a separate bowl and set aside.
Add 1 cup of the flour along with all of the baking powder, baking soda, and salt, mixing just until combined. Add ½ of the liquids buttermilk mixture, then repeat, alternating with the remaining flour and liquids to form a fairly thick batter.
Generously spray a 10- or 12-cup bundt pan with baking spray or grease and flour the pan, making sure to get all of the crevices. Pour the batter into the pan, smoothing it out on top and tap it against the counter a few times to knock out air bubbles.
Bake for 55 to 70 minutes until a wooden skewer inserted into the center of the cake comes out clean with just a few crumbs clinging to it. Cool in the pan for 10 to 15 minutes, then turn out onto a cake plate or wire rack.
Make the lemon syrup by combining the lemon juice and sugar in a microwave-safe bowl. Microwave for 30 seconds, then stir to dissolve the sugar. Heat for another 30 seconds if needed to dissolve all of the sugar.
Brush the glaze over the still-hot cake using a pastry brush until all of most of it has been absorbed. Cool completely. At this point the cake can be wrapped in plastic wrap and refrigerated for up to 5 days or frozen for up to 3 months.
Make the icing by whisking the powdered sugar and lemon juice together. The icing should be thick but still pourable. You can add more lemon juice or powdered sugar as needed to get the right consistency. Pour over the cake and serve once the icing has set, about 30 minutes.