Make the Cookie Dough: In a large bowl, beat the butter, granulated sugar, and brown sugar together on medium-high speed until fluffy (2–3 minutes). Add the egg and vanilla extract, and beat on high until combined.
In a separate bowl, whisk together the flour, cocoa powder, hot cocoa mix, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients and mix on low speed until combined. Add milk and mix. Dough will be thick and sticky.
Cover dough tightly and chill for at least 2 hours, or up to 3 days.
Bake the Cookies: Preheat oven to 350°F (177°C). Line baking sheets with parchment or silicone mats. Roll chilled dough into balls (~25g each) and arrange 2–3 inches apart. Bake for 10 minutes.
Remove cookies from oven, lightly press a marshmallow half into each cookie, and return to the oven for 2 minutes. Press down on the marshmallows gently to flatten.
Cool cookies on the baking sheet for 10 minutes, then transfer to a wire rack.
Melt the Chocolate: Melt the chocolate in a microwave or double boiler until smooth. Drizzle or spoon the melted chocolate over the marshmallow-topped cookies. Let set at room temperature for 30–60 minutes.
Store leftover cookies tightly covered at room temperature for up to 1 week.