In a medium saucepan, melt the butter over medium heat, stirring constantly, until it turns golden brown and has a nutty aroma. This should take about 5 minutes. Let it cool slightly.
Add the brown sugar to the melted brown butter and stir continuously until dissolved, creating a caramel-like flavor.
In a large bowl, whip the cold heavy cream until stiff peaks form. It's important that the cream is very cold, otherwise, it won't whip properly.
Gently fold in the sweetened condensed milk, vanilla extract, and a pinch of salt into the whipped cream, being careful not to deflate the cream.
Gradually add the prepared brown butter to the cream mixture while continuously stirring, ensuring the butter is evenly distributed throughout the ice cream.
Pour the mixture into a deep container, cover, and freeze for at least 6 hours, or preferably overnight, to allow it to set.
Before serving, let it sit at room temperature for 5-10 minutes to soften slightly and make it easier to scoop. Garnish with extra browned butter and a sprinkle of brown sugar, if desired.
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