If using real mascarpone cheese, in a mason jar or other airtight container, whisk together mascarpone and vanilla syrup until completely smooth. Add milk and cream to the container and shake vigorously to combine.
If using instant pudding mix, simply add milk, cream, vanilla syrup, and pudding mix to a jar or other airtight container and shake vigorously to combine.
This is your mascarpone cold cream... Keep it in the fridge until you're ready to turn it into cold foam right before using.
To turn into foam, you can either use an electric frother, hand mixer, or blender with about ⅓ cup of the cold cream at a time. Alternatively, you can shake the mixture vigorously in a mason jar until it's frothy.