Blackberry–Cheesecake Swirl Ice Cream

HOW TO MAKE IT :

1. Combine the blackberries, . cup granulated sugar and the lemon juice in a medium saucepan and toss well. Let stand until the juices start to pool, about 10 minutes. Mash with a potato masher until the berries are broken down, then cook over low heat until the berries are very soft and the juices are slightly thickened, about 20 minutes.
2. Pour the mixture through a fine-mesh strainer into a medium bowl, pressing with a spoon to force out the liquid. Set the liquid aside to cool. Discard the pulp and seeds. 
3. Heat the half-and-half and remaining 1 cup granulated sugar in a large saucepan over low heat, stirring occasionally, until just simmering. Remove from the heat.
4. Beat the egg yolks in a large bowl with a mixer on medium-high speed until pale yellow and thick, 3 to 5 minutes. Temper the egg yolks by slowly drizzling in ¾ cup of the hot half-and-half mixture, whisking constantly. Pour the tempered egg yolk mixture into the saucepan with the rest of the half-and-half mixture, whisking gently as you add it. Simmer over low heat, stirring constantly with a wooden spoon, until very thick, about 5 minutes. The mixture should be thick enough to coat the back of the spoon but not as thick as pudding.
5. Pour the mixture through a fine-mesh strainer into a separate large bowl. Add the heavy cream and blackberry syrup; stir to combine. Refrigerate until chilled, at least 4 hours or overnight.
6. Churn the mixture in an ice cream maker according to the manufacturer’s directions. When it’s almost ready, make the cheesecake mixture: Beat the cream cheese, lemon zest and powdered sugar in a large bowl with a mixer on low speed until combined, about 1 minute. Increase the speed to medium high and beat until fluffy and very soft, 1 to 2 more minutes.
7. Transfer half the ice cream to a freezer-safe baking dish or loaf pan. Spoon half the cheesecake mixture over the top and gently swirl with a butter knife. Repeat with the remaining ice cream and cheesecake mixture. Cover and freeze until firm enough to scoop, at least 8 hours.