1. Make the ice cream base by combining cream cheese, sugar, and heavy cream in a large saucepan over medium heat. Cook, stirring often, until cream cheese is melted and the ingredients become one smooth mixture. I like to use a whisk to really cream them together. If your cream cheese is cold this will take longer.
2. Whisk egg yolks with half and half and add to the hot cream cheese mixture, whisking vigorously as you add. Cook this mixture, stirring often, over medium heat until it thickens slightly and you see little wisps of steam. Do not boil. If you have a thermometer, you are looking for a temperature of 170 degrees F.
3. Add vanilla, stir, and transfer to a large bowl or measuring cup and place in refrigerator until completely chilled. Feel free to strain through a fine mesh strainer if you feel you may have curdled anything, but I almost never bother with this step in my ice cream recipes anymore. 
4. Make the graham cracker mixture by melting butter in a skillet and adding graham cracker crumbs and sugar. Stir often over medium heat until the crumbs brown slightly and the mixture becomes fragrant. Let cool to room temperature and store in an airtight container until ready to use. 
5. Wipe out the graham cracker pan and use it to make the blackberry sauce. Combine berries, lemon juice, sugar, and cornstarch in skillet over medium high heat. Cook until berries are very soft and sugar is completely dissolved and creates a syrup with the berry juices. This should all take about 5 minutes.
6. Mash berries with a fork or potato masher before straining the mixture through a sieve. Use a wooden spoon to really mash them through. You should end up with about 8 ounces. Allow berry sauce to cool completely.

Bring it all together:

Place a 2 quart container in the freezer. I often use a 9x5 inch loaf pan. When the ice cream base is cold, add to ice cream maker along with lemon juice and churn according to your ice cream maker instructions. When ice cream is done, transfer 1/3 of mixture to frozen container. Add 1/3 of blackberry sauce by dropping spoonfuls here and there over ice cream. Then add 1/3 of graham cracker crumbs in the same manner, followed by another layer of ice cream. Continue until you have completed 3 layers. Take a chopstick or butter knife and poke all the way to the bottom and make trails all over ice cream keeping the chopstick on the bottom the whole time. 

When you are done swirling, cover ice cream with plastic wrap and place in freezer until firm - about 6-8 hours.