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Preheat the oven to 350°F.
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Beat together the butter, sugar, and salt, first until combined, then until fluffy and lightened in color. For a visual of what this should look like, see our video, how to cream butter and sugar.
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Add the eggs one at a time, beating well after each addition. Scrape the sides and bottom of the bowl once all the eggs have been added, and beat briefly to re-combine any residue.
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Measure the flour by gently spooning it into a cup, then sweeping off any excess. Whisk the baking powder into the flour. Add the flour mixture to the batter in three parts alternately with the milk, starting and ending with the flour. The batter may look slightly curdled when you add the milk. That's OK; it'll smooth out as you add the flour. Mix until everything is well combined; the batter will look a bit rough, but shouldn't have any large lumps. Stir in the zest or lemon oil.
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Thoroughly grease a 10- to 12-cup Bundt pan. Spoon the batter into the prepared pan, leveling it and smoothing the top with a spatula.
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Bake the cake for 45 to 60 minutes, or until a cake tester or toothpick inserted into the center comes out clean. A pan with a dark interior will bake cake more quickly; start checking at 40 minutes.
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While the cake is baking, make the glaze by stirring together the lemon juice and sugar. Microwave or heat over a burner briefly, stirring to dissolve the sugar. You don’t want to cook the lemon juice, so microwave just until very warm, but not uncomfortably hot — less than 1 minute should do it. Set the glaze aside.
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Remove the cake from the oven, and carefully run a knife between cake and pan all around the edge. Place the pan upside down on a cooling rack. If the cake drops out of the pan onto the rack, remove the pan. If the cake doesn’t drop onto the rack, let it rest for 5 minutes, then carefully lift the pan off the cake. If the cake still feels like it's sticking, give it another 5 minutes upside down, then very gently shake the pan back and forth to loosen and remove it.
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Brush the glaze all over the hot cake, both top and sides. Let it sink in, then brush on more glaze, continuing until all the glaze is used up.
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Allow the cake to cool completely before icing and serving.
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To ice the cake, mix the sugar and salt, then mix in 2 tablespoons of the lemon juice, adding just enough additional juice to create a thick glaze, one that's just barely pourable. Drizzle it artfully over the completely cool cake.
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Storage information: Store the cake, well wrapped, at room temperature for several days. Freeze for longer storage.
