Add the butter to a microwave-safe mug and microwave for 20-30 seconds until fully melted.
Add both sugars, vanilla extract, and egg yolk to the mug. Stir well with a spoon until fully combined.
In a separate small bowl, mix the flour, cocoa powder, and salt. Add to the mug and stir until just combined and smooth.
Pour in the milk to adjust thickness and stir gently. The batter should be thick but pourable.
Microwave on high for 90 seconds. Check for doneness; if still undercooked, 15-second intervals until done. Be careful not to overcook.
Let cool for 1-2 minutes before enjoying directly from the mug.
Notes
For an extra decadent treat, top with a scoop of vanilla ice cream and chocolate syrup. Store leftovers covered at room temperature for up to 24 hours.