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Preheat the oven to 350°F with a rack in the center. Line a metal 8" square pan with a parchment sling and lightly grease.
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To make the bars: In a medium bowl, combine the sugar, zest, salt, and vanilla.
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In a small saucepan or in a microwave-safe bowl, melt the butter and lemon juice, then pour it directly over the sugar mixture.
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Whisk gently to combine, then mix vigorously for 30 seconds. The mixture will become thick and shiny as it emulsifies. Add the eggs and continue whisking until just combined.
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Using a flexible spatula, mix in the flour and baking powder until no streaks of flour remain.
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Transfer the batter to the prepared pan and spread it evenly to the edges.
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Bake the bars for 28 to 32 minutes until the center is set, the bars are golden, and a toothpick or thin knife inserted into the center comes out with a few moist crumbs. It’s OK if the toothpick or knife isn’t clean, but you don’t want to see any raw batter.
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Remove the bars from the oven and cool them completely in the pan (about 1 hour) before frosting.
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To make the strawberry frosting: Place the freeze-dried strawberries in a small ziplock bag and use a rolling pin or the bottom of a measuring cup to crush them into a fine powder. Once ground, you should have about 1/4 cup (12g) of powder.
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In a medium bowl, combine the freeze-dried strawberry powder, butter, and salt and stir with a flexible spatula until thoroughly combined.
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Add the confectioners’ sugar, vanilla, and milk and continue mixing gently with the spatula until combined. Once the confectioners’ sugar is incorporated, stir vigorously with the spatula to create a light and fluffy texture. (The mixture will look dry at first but after a couple of minutes, it will become fully combined.)
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With a small offset spatula, spread the frosting over the top of the cooled bars. Slice and serve.
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Storage information: Store any leftover strawberry lemonade bars, covered, at room temperature for several days; freeze for longer storage.
