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To make the cake: Preheat the oven to 350°F with a rack in the center. Lightly grease a tea loaf pan or 9" round cake pan, then line with parchment. (See “tips,” below for more information on pan size.)
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In a stand mixer fitted with the flat beater attachment, or working with an electric hand mixer in a large bowl, beat (“cream”) together the butter, sugar, and lemon zest until light and fluffy, 3 to 5 minutes.
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Meanwhile, in a medium bowl or liquid measuring cup, whisk the yogurt, sourdough discard, lemon juice, and vanilla until homogenous. In a separate medium bowl, whisk the flour, baking powder, salt, and baking soda. Set both bowls aside.
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Add the eggs to the creamed butter mixture one at a time, beating until fully combined and scraping down the bowl between each addition.
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Add about one-third of the flour mixture to the bowl, stirring on low to combine, then add about half of the yogurt mixture. (It’s OK if the batter looks slightly curdled once the liquid goes in.) Add another third of the flour mixture, followed by the remaining yogurt mixture. Finish by stirring in the remaining flour mixture. Scrape down the sides of the bowl, then mix on medium-low speed for 15 to 30 seconds to ensure the batter is homogenous and smooth.
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Transfer the batter to the prepared pan and spread into an even layer.
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If baking a tea loaf, bake for 40 to 45 minutes, tenting with foil if necessary to prevent overbrowning, until a toothpick or paring knife comes out clean and the internal temperature registers at least 190°F on a digital thermometer. If baking a round, bake for 35 to 40 minutes, until set; the cake should spring back when lightly pressed. Let the cake cool in the pan on a wire rack for 10 to 15 minutes, until cool enough to handle, before transferring directly to the wire rack to cool completely.
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To make the glaze: In a small bowl, stir or whisk together all the glaze ingredients to combine; the glaze will be slightly thicker than honey. Add lemon juice by the 1/2-teaspoon, if necessary, to achieve the right consistency. Pour the glaze over the cooled cake, then use a spoon or small offset spatula to encourage it to drip down the sides. Garnish with lemon zest, if desired. Slice and serve at room temperature.
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Storage information: Store any leftover sourdough lemon cake, covered, at room temperature for up to 3 days; freeze cake, unglazed, for longer storage.
