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Preheat the oven to 350°F. Grease an 8” square pan, line it with parchment, and grease the parchment.
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To make the crust: In a medium bowl, beat the cookie mix and butter until crumbly.
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Add the egg and water and beat until the mixture becomes a cohesive dough.
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Transfer the dough to the prepared pan. Evenly press it into the bottom of the pan; your wet fingers work well here.
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Bake the crust for 12 to 16 minutes, until it’s golden brown; remove it from the oven. Reduce the oven temperature to 325°F.
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To make the filling: In a large bowl, mix the cream cheese and sugar at low speed until smooth and lump-free. Add the heavy or sour cream, eggs, vanilla, lemon zest, and juice, stirring until thoroughly combined. Pour the filling into the crust.
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To make the swirl: Place the berries and sugar in a small saucepan or microwave-safe bowl and cook, stirring occasionally, until the berries collapse and the sugar dissolves, about 5 minutes. Strain the mixture, reserving the sweetened juice and discarding the solids.
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To add the swirl: Take about 1/3 cup of the sweetened juice and pour it over the filling in lines about 1” apart. Use a table knife to swirl it into the filling. Save the remaining juice for serving or another use.
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Bake the bars for 30 to 40 minutes, until the filling is set at the edges but still wobbly in the center. A digital thermometer will read 180°F when inserted into the filling 1” in from the edge.
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Remove the bars from the oven and cool to room temperature; refrigerate until ready to serve.
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Store any leftover bars, well wrapped, in the refrigerator for up to a week; freeze for longer storage.