Crush the cereal in a food processor, or by hand, until finely ground.
Combine the crushed cereal with the brown sugar and 2 cups of the heavy cream, reserving 1 cup of heavy cream for later. Soak the cereal for 30 minutes.
Once soaked, set up a bowl with a strainer and 4 layers of cheesecloth. Pour the cereal mixture into the cheesecloth to strain into the bowl. Squeeze any additional cream from the cereal by bunching together the loose ends of the cheesecloth, turning it into a pouch holding the cereal, and squeezing excess liquid into the strainer.
Set the soaked cereal aside (optional)
Combine the strained milk and the reserved cup of heavy cream in a large bowl or stand mixer. Whip the heavy cream mixture on high for 2-3 minutes. The whip cream is the right consistency when stiff peaks are formed.
Pour the sweetened condensed milk and vanilla extract into the whipped cream. Whip on low for 30 seconds.
If desired, gently mix in the leftover soaked cereal.
Pour the ice cream mixture into an airtight and freezer safe container. Press a layer of plastic wrap or parchment paper directly onto the ice cream.
Cover the container with an airtight lid and freeze for 2 hours for a soft serve consistency, or at least 4 hours for a scoopable consistency.