Mix the softened cream cheese and honey in a medium mixing bowl until well combined. Add in the milk, almond extract, fresh lemon zest, and salt, and combine. Finally, add the heavy cream and mix until well-combined.
Cover the bowl with plastic wrap and place in the refrigerator to chill for at least 2 hours (although you can chill it overnight).
Meanwhile, combine the fresh blackberries, honey, and pinch of salt in a small saucepan over medium heat. Stir frequently until the blackberry mixture begins to simmer. Use a fork or potato masher to mash the berries.
Continue to simmer for about 10 minutes, or until the blackberry sauce as slightly thickened and reduced in volume.
Place the blackberry mixture in a glass container and store in the refrigerator until ready to churn.
When ready to churn, pour the cheesecake ice cream mixture into your chilled glass dish or bread pan.
Then take the blackberry sauce and spoon it onto the cheesecake ice cream base in the shape of a cross (for a round dish) or in several parallel lines (for a bread pan).
Use a butter knife to make a circle, swirling the blackberry mixture into the ice cream mixture, cover, and place in the freezer.
Stir the ice mixture every 30-45 minutes, scooping the ice cream along the edge into the center to allow everything to freeze evenly.
Freeze for an additional 4-5 hours.