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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 raspberry layer: In a small saucepan or pot, combine the raspberries, sugar, and lemon juice. Cook over medium heat, stirring frequently, until the mixture comes to a rolling boil, and the raspberries break down and release their juice.
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Boil the raspberry mixture for 1 minute, then remove from the heat and stir in the flour; the mixture will thicken considerably. Set aside to cool to room temperature while you prepare the rest of the cake.
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To make the cream cheese layer: In a small bowl, use a flexible spatula to mix the cream cheese and sugar together until smooth.
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Add the raspberries and mix until incorporated. The berries will break down slightly; that’s OK. Set aside.
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To make the crumbs: In a medium bowl, whisk together the flour, sugars, and salt. Add the butter, cream cheese, and vanilla; toss to combine. Using a bowl scraper, fork, or your fingers, work in the butter and cream cheese until crumbs form and no dry spots remain. The crumbs should be irregular, ranging in size from a pea to walnut. Set aside.
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To make the cake: In a large bowl or bowl of a stand mixer, combine the flour, sugar, baking powder, salt, baking soda, and lemon zest. Add the butter and mix at low speed using an electric hand mixer or the flat beater on a 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 vanilla and egg, 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 low speed for 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 cake: Transfer the batter into the prepared pan, spreading it all the way to the edges. (A small offset spatula is a helpful tool here.)
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Dollop the raspberry layer across the batter and spread evenly.
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Dollop heaping teaspoons of the cream cheese layer on top of the raspberry layer; a teaspoon cookie scoop works well here. Do not spread.
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Sprinkle the crumbs evenly over the cream cheese layer and gently press to adhere.
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Bake the cream cheese coffee cake for about 35 to 40 minutes, until it's set in the middle and the crumbs are lightly browned. (Check for doneness after 35 minutes and add additional time only as needed to avoid overbaking the cake.) It's done when a toothpick or paring knife inserted into the center comes out with just a few moist crumbs clinging to it.
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Remove the coffee cake from the oven and allow it to cool in the pan for 30 minutes.
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Remove the coffee cake from the pan using the overhanging edges of parchment as handles to gently lift it out. Place the coffee cake (still on the parchment or pan liner) on a wire rack to cool slightly.
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To make the topping: Using a small food processor or blender, pulse the freeze-dried raspberries and confectioners’ sugar together until they’re a fine powder. Use a fine-mesh strainer to sprinkle the topping over the coffee cake before slicing and serving.
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Storage information: Store the cream cheese coffee cake, well wrapped, at room temperature for several days; freeze for longer storage.
