To make the Cinnamon Cream Cheese Ice Cream:

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1. In a medium saucepan, combine the milk, cream, and sugar over medium heat, stirring often to dissolve the sugar.
2. In a small bowl, beat the eggs.
3. Once the milk mixture is steaming, but not yet at a boil, slowly pour about 1/4 cup (60 milliliters) into the eggs while constantly whisking.
4. Once incorporated, slowly stream in and whisk another 1/2 cup (125 milliliters). Slowly pour the mixture back into the saucepan over medium-low heat while whisking. Cook while continuing to stir until just thickened enough to coat the back of a spoon, 170-180˚F (77˚C-82˚C). Be careful not to overcook or it will curdle.
5. Remove from heat and pour through a fine mesh strainer into a large bowl. Whisk in the diced cream cheese until melted and smooth, then stir in the cinnamon and vanilla.
6. Cover and refrigerate until well chilled, about 4 hours to overnight.

To make the Cinnamon Swirl:

1. In a small saucepan, combine the brown sugar, water, butter, and cinnamon over medium heat. Cook, constantly stirring, until the butter has melted and the sugar has completely dissolved.
2. In a small bowl, whisk together the cornstarch and water. Pour into the simmering cinnamon sauce and cook, stirring constantly, until the mixture thickens.
3. Remove from heat and stir in the vanilla extract. Refrigerate the cinnamon mixture in an airtight container until well chilled, about 2-4 hours.

To assemble the ice cream:

1. Transfer the chilled cinnamon cream cheese mixture to an ice cream machine and churn according to manufacturer's instructions until thickened and creamy.
2. In a large, freezer safe container, coat the bottom with a small layer of the chilled cinnamon swirl. Add a layer of churned ice cream, more cinnamon sauce, and continue layering as desired, ending with a top layer of ice cream and a drizzle of the sauce. Do not stir or mix. Cover and freeze until firm, at least 4 hours.
3. If any of the cinnamon sauce is leftover, refrigerate and save to use as a topping.