Cream the butter and sugars: In a large mixing bowl, beat ½ cup softened unsalted butter, ½ cup packed light brown sugar, and 2 tablespoons granulated sugar for about 2 minutes, until light and fluffy. Add 1 large room-temperature egg and 1 teaspoon vanilla extract, then beat until smooth and combined.
Add the dry ingredients: In a separate bowl, whisk together 1 cup all-purpose flour, ½ cup unsweetened cocoa powder, baking soda, and salt. Gradually add the dry ingredients to the wet mixture and mix on low speed just until a soft chocolate cookie dough forms. Avoid overmixing.
Chill the cookie dough: Cover the bowl and refrigerate the dough for 30 minutes. Chilling helps these hot chocolate cookies stay thick, soft, and fudgy instead of spreading too much in the oven.
Shape the cookies: Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper. Scoop about 2 tablespoons of dough per cookie, roll into balls, and arrange them about 3 inches apart. Gently flatten each ball to about ½-inch thick.
Bake and add the chocolate: Bake for 10 minutes, or until the edges are just set while the centers remain soft. Remove the baking sheet from the oven and spoon about 1 tablespoon of melted chocolate over each cookie. Gently spread the chocolate, then press one marshmallow half, cut-side down, into the center.
Finish baking: Return the cookies to the oven and bake for another 2 minutes, until the marshmallows are puffed and slightly melted. For toasted marshmallow tops, broil for 1–2 minutes, watching closely to prevent burning.
Cool: Let the cookies cool on the baking sheet for 10 minutes before transferring them to a wire rack. The cookies will be very soft when hot but will firm up as they cool while staying soft and chewy in the center