You have probably been making hot chocolate without a recipe for your entire life — for many, hot chocolate can mean dumping a few scoops of a mix into a mug, adding a few glugs of milk, and heating the whole thing in the microwave. Making hot chocolate from scratch was something you probably didn't even think to do. With a just a few simple steps, you'll realize how truly simple it is. Hot chocolate is like pancakes: Both aren't hard to make from scratch, but most people use mixes and the end results suffer.
Heat milk and half-and-half in a saucepan over medium heat to just below the simmering point. Remove the pan from the heat and add both types of chocolate. When chocolate has melted, add the sugar and vanilla and whisk to combine. Reheat gently and serve immediately garnished with whipped cream and grated chocolate.