In a large bowl, cream together cream cheese and butter until fully incorporated and fluffy. Add heavy cream, sweetened condensed milk, and vanilla extract and stir until combined and smooth.
Pour into ice cream maker and churn according to the manufacturer's instructions, about 20-30 minutes.
About 5 minutes before the ice cream is done, add in chunks of approximately 3-4 cupcakes. Finish churning for an additional 5-10 minutes
When finished, scoop mixture into resealable, freezer safe container. Gently fold in another 3-4 cupcake chunks.