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1
Spread the cookie butter on a sheet of parchment paper using an offset or rubber spatula, shaping it into a roughly 10-by-6-inch rectangle.
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2
Transfer the parchment to a baking sheet and place it in the freezer until the cookie butter is stiff and pieces easily tear off, at least 2 hours.
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3
(It will not be completely frozen.)
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4
Meanwhile, place the strawberries and 1/3 cup of the sugar in a small, nonreactive saucepan and stir to coat the strawberries with the sugar.
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5
Mash with a potato masher until about half of the strawberries are completely smashed but some medium-sized chunks remain, about 1 minute.
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6
Place the pan over high heat and cook, stirring occasionally, until bubbles form along the edge, about 3 minutes.
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7
Skim any foam from the surface of the sauce with a spoon and discard.
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8
Add 1/2 teaspoon of the lemon zest and 1 1/2 teaspoons of the lemon juice and stir to combine.
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9
Bring to a full boil, cooking until foam coats the surface, about 1 minute.
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10
Remove from the heat and skim and discard the foam.
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11
Transfer the sauce to a medium heatproof bowl and let it cool to room temperature, about 45 minutes.
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12
Cover and refrigerate until completely chilled, at least 1 hour.
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13
When the cookie butter and strawberries are ready, place the remaining teaspoon of lemon zest, the remaining tablespoon of lemon juice, the remaining 2/3 cup of sugar, the milk, and the cream cheese in a blender and blend until smooth.
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14
Immediately freeze in an ice cream maker according to the manufacturers directions.
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15
Keep the ice cream in the bowl of the ice cream maker.
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16
Remove the cookie butter from the freezer, tear it into rough 1- to 2-inch pieces, and fold them into the finished ice cream with a rubber spatula.
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17
(Work quickly, as the cookie butter will soften and be difficult to handle as it warms up.)
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18
Add the chilled strawberry sauce and fold just until a thick swirl is visible.
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19
Serve immediately for a soft ice cream, or transfer the ice cream to an airtight container and freeze until solid.
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20
Allow the ice cream to sit at room temperature for 15 minutes before serving if frozen solid.
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21
The ice cream can be kept for up to 2 weeks.
