Start by beating the butter on medium speed until it becomes smooth and fluffy. Make sure to scrape the sides of the bowl and the beaters a few times to ensure there are no lumps in the butter. You can use either a hand mixer or the paddle attachment on your stand mixer for this step.
Next, add a couple of tablespoons of cold sweetened condensed milk and the vanilla to the beaten butter. Continue beating until they are well incorporated.
Now, add the remaining sweetened condensed milk in small portions, making sure to mix well after each addition.
Once you've added all the condensed milk, increase the speed to medium-high and continue beating for at least two minutes or until the frosting becomes smooth and airy.