Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper. Separate the cinnamon rolls and arrange them on the prepared baking sheet.

Bake for 12-15 minutes, or according to package directions, until golden brown and cooked through. Let them cool slightly on the baking sheet.

While still warm, drizzle the reserved icing from the can over the baked cinnamon rolls. Allow them to cool completely. Once fully cooled, chop the cinnamon buns into small, bite-sized pieces (about 1/2 to 3/4 inch). Place the chopped pieces in a freezer-safe bag or container and freeze for at least 30 minutes, or until solid.

In a medium saucepan, melt the unsalted butter over medium heat.

Add the brown sugar, ground cinnamon, and salt to the saucepan. Whisk until smooth. Bring the mixture to a gentle simmer and cook for 2-3 minutes, stirring occasionally, until it thickens slightly.

Remove the saucepan from the heat. Stir in the heavy cream and vanilla extract. Transfer the cinnamon swirl to a heatproof bowl and let it cool completely to room temperature. It will thicken further as it cools.

In a large mixing bowl, whisk together the heavy cream, whole milk, granulated sugar, and a pinch of salt until the sugar is completely dissolved.

In a separate small bowl, whisk the softened cream cheese until it is smooth and lump-free. Gradually whisk in about 1/2 cup of the cream mixture into the cream cheese until it's fully combined and smooth. This step tempers the cream cheese, preventing lumps.

Pour the cream cheese mixture back into the large bowl with the rest of the cream mixture. Whisk thoroughly until everything is well incorporated and smooth. Stir in the vanilla extract.

Cover the bowl and refrigerate the ice cream base for at least 4 hours, or preferably overnight, until it is thoroughly chilled. This is crucial for a smooth ice cream texture.

Once the ice cream base is thoroughly chilled, churn it in your ice cream maker according to the manufacturer's instructions, typically for 20-25 minutes, until it reaches the consistency of soft-serve ice cream.

During the last 5 minutes of churning, add the frozen, chopped cinnamon bun pieces to the ice cream maker. Continue churning until they are evenly distributed.

Transfer half of the churned ice cream to a freezer-safe container. Drizzle about half of the cooled cinnamon swirl over the ice cream. Use a spoon or knife to gently swirl it into the ice cream.

Add the remaining churned ice cream to the container, then drizzle with the rest of the cinnamon swirl. Swirl again gently to create marble patterns.

Cover the container tightly and freeze for at least 4-6 hours, or until the ice cream is firm enough to scoop, before serving.