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Preheat the oven to 350°F. Prepare an 8" square pan with a parchment sling or reusable silicone liner, then lightly grease the sides of the pan. Alternatively, line a 9" cake pan with a round piece of parchment or round reusable liner and lightly grease the sides.
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To make the crumbs: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. In a medium bowl, whisk together the flour, sugars, salt, and spices. Add the butter and vanilla; toss to combine. Using a bowl scraper, a fork, or your fingers, work in the butter until crumbs form and no dry spots remain. The crumbs should be irregular, ranging in size from a pea to walnut.
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Add the diced ginger and toss until incorporated. Set aside.
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To make the cake: In a large bowl or bowl of a stand mixer, combine the flour, sugar, lemon zest, baking powder, salt, baking soda, and spices. Add the butter and mix at low speed using an electric hand mixer or the flat beater attachment of the stand mixer, until the mixture looks sandy, about 1 to 2 minutes. (This mixing method is called reverse creaming and is also known as the paste method.)
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Beat in the egg, lemon paste, and vanilla, then mix until no dry spots of flour remain. Stop the mixer and scrape the bowl and beater(s) as needed; the mixture will be thick.
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Add the yogurt or sour cream all at once before starting to mix. Stir until the batter is smooth. Stop to scrape the bowl and then mix again at medium-low speed for 30 seconds to 1 minute; this final mix is key to ensuring the best texture so don’t skip it.
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To assemble and bake the lemon coffee cake: Transfer the batter to the prepared pan and spread it evenly all the way to the edges. (A small offset spatula is a helpful tool here.)
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Spread the lemon curd evenly over the top of the batter.
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Sprinkle the crumbs evenly over the lemon curd and gently press to adhere.
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Bake the lemon coffee cake for 42 to 46 minutes, until it's set in the middle and the crumbs are golden brown. The middle of the cake will be firm to the touch with no evidence of uncooked batter. (Use these cues for doneness rather than testing with a toothpick or paring knife, as the lemon curd will remain fluid and may look like uncooked batter.)
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Remove the lemon coffee cake from the oven and allow it to cool in the pan for 20 minutes. If making the glaze, prepare it while the coffee cake is cooling.
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To make the glaze: In a small bowl, stir together the glaze ingredients until smooth, starting with smaller amount of water. Add additional water as needed until the glaze is the consistency of molasses.
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Remove the lemon coffee cake from the pan using the overhanging edges of parchment as handles to gently lift it out. (If you used a silicone liner or parchment round, run a nylon spreader or dull knife around the edges of the pan to loosen the cake and then gently tip it out.) Place the lemon coffee cake, still on the parchment or liner, on a wire rack to cool slightly.
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Drizzle the glaze over the lemon coffee cake, or finish by dusting with non-melting sugar or confectioners’ sugar.
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Storage information: Store leftover lemon coffee cake, well wrapped, at room temperature for several days; freeze for longer storage.