Strawberry Cheesecake Ice Cream

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.

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. (It will not be completely frozen.)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. Mash with a potato masher until about half of the strawberries are completely smashed but some medium-sized chunks remain, about 1 minute.

Place the pan over high heat and cook, stirring occasionally, until bubbles form along the edge, about 3 minutes. Skim any foam from the surface of the sauce with a spoon and discard.

Add 1/2 teaspoon of the lemon zest and 1 1/2 teaspoons of the lemon juice and stir to combine. Bring to a full boil, cooking until foam coats the surface, about 1 minute.

Remove from the heat and skim and discard the foam.

Transfer the sauce to a medium heatproof bowl and let it cool to room temperature, about 45 minutes. Cover and refrigerate until completely chilled, at least 1 hour.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. Immediately freeze in an ice cream maker according to the manufacturer’s directions. Keep the ice cream in the bowl of the ice cream maker.

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. (Work quickly, as the cookie butter will soften and be difficult to handle as it warms up.)

Add the chilled strawberry sauce and fold just until a thick swirl is visible.

Serve immediately for a soft ice cream, or transfer the ice cream to an airtight container and freeze until solid. Allow the ice cream to sit at room temperature for 15 minutes before serving if frozen solid. The ice cream can be kept for up to 2 weeks.