In a medium saucepan, stir granulated sugar, brown sugar, and cocoa powder together.
Add heavy cream and semi-sweet chocolate and heat over medium heat, stirring continuously until the chocolate has melted and the mixture just begins to bubble (but before it turns to a full boil). Reduce to low heat and simmer for 3 minutes, stirring frequently.
Add butter and stir until the butter has melted. Remove from heat and stir in the vanilla extract.
Set aside to cool, then transfer to a mason jar or airtight container. Store in the refrigerator for up to 30 days.
To Serve: Portion the desired amount into a microwave-safe bowl. Heat in 15-second increments stirring after each time, until the sauce is warm.