Pour the whole milk and heavy cream into a saucepan and place over medium-low heat. Slowly warm the mixture until hot but not boiling.

While the milk warms, sift together cocoa powder, powdered sugar, and optional cornstarch into a small bowl to break up any lumps.

When the milk is hot, whisk in the sifted cocoa mixture. Continue heating over medium-low, whisking constantly for about 5 minutes until the mixture thickens slightly.

Stir in the chopped dark chocolate pieces and whisk until fully melted. Adjust sweetness by adding more powdered sugar if needed. Remove from heat.

Pour the hot chocolate into two mugs. Top with whipped cream and sprinkle with dark chocolate shavings.

Chill the mixing bowl and whisk attachment in the freezer for at least 30 minutes.

Pour the cold heavy cream into the chilled bowl and whisk on high speed until soft peaks form. Gradually add powdered sugar and vanilla extract, whisking until medium peaks form.