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4 cups whole or low-fat milk

1/2 teaspoon pure vanilla extract

3/4 cup sugar

1/2 cup unsweetened cocoa powder

4 ounces semisweet or bittersweet chocolate, finely chopped

3 tablespoons crushed peppermint sticks, plus more for serving

Pinch kosher salt

Unsweetened whipped cream, for serving

Peppermint sticks, for serving