In a small saucepan, melt butter over medium heat until golden and nutty to form a brown-butter base.
Add brown sugar to the melted butter and stir until dissolved and syrupy.
Pour in heavy cream and milk, whisking to combine; warm gently over low heat without boiling.
Stir in butterscotch or caramel sauce, vanilla extract, and a pinch of salt; heat until steaming, then remove from heat.
For hot butterbeer: divide the warm base among four mugs and top each with warmed cream soda or warmed milk mixed with simple syrup.
For cold butterbeer: chill the base, then combine equal parts chilled base and cream soda in glasses over ice.
Make the foam: whip 2 tbsp heavy cream with 1 tbsp butterscotch to soft peaks (or use stabilized whipped cream).
Spoon a generous dollop of foam onto each drink and top with mini or toasted marshmallows.
Drizzle extra caramel over the foam and sprinkle crushed shortbread for texture, if desired.
Serve immediately—add a splash of spiced rum or butterscotch schnapps if making an adult version.