Place chopped dark chocolate in a heatproof bowl.

In a saucepan, whisk milk, sugar, cocoa powder, cornstarch, and salt. Cook over medium heat, whisking constantly, until thickened and bubbling (10 minutes).

Remove saucepan from heat and pour mixture through a fine mesh strainer into the bowl with chocolate.

Let sit for a minute, then whisk until smooth and combined.

Stir in vanilla extract and butter until melted and incorporated.

Divide mixture evenly among 4-6 serving dishes. Cover with plastic wrap, touching the surface of the budino.

Refrigerate for at least 2 hours, or until set.

Remove plastic wrap, add optional toppings, and serve.