Place 1 cinnamon stick in each of 4 small mugs.
Whisk both milks in a saucepan until combined. Cook over medium-high heat, stirring occasionally, until just beginning to emit steam, 5 to 6 minutes. Combine cocoa and cinnamon in an empty pot or large liquid measuring cup and whisk in 1/4 cup milk mixture until smooth. Slowly drizzle in remaining hot milk mixture, whisking constantly. Whisk in hot coffee. Divide among prepared mugs and serve immediately.
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