Prepare the pan and oven
Preheat the oven to 350°F (175°C). Line an 8×8 inch (20×20 cm) baking pan with parchment paper or lightly grease it, making sure you can lift the brownies out easily after baking.​

Melt the butter and chocolate
In a microwave-safe bowl, combine the 1/2 cup butter and 6 oz chocolate chips. Microwave in 30-second bursts, stirring between each, until the chocolate is fully melted and smooth (about 90 seconds total). Let the mixture cool for 2 minutes so it’s warm but not hot.​

Whip the eggs and sugar
In a medium mixing bowl, add the 1 cup sugar and 3 large eggs. Use a hand mixer or stand mixer on high speed to whip for about 3 minutes, until the mixture is thickened, lighter in color, and slightly foamy. This step helps create that shiny, crackly top.​

Combine the wet ingredients
Pour the melted chocolate mixture into the egg mixture and gently fold with a spatula until fully combined. Add the vanilla extract and stir again.​

Add the dry ingredients
Sift or whisk together the flour, cocoa powder, and salt in a small bowl. Add this mixture to the wet ingredients and fold just until no dry streaks remain. The batter will be thick and glossy; avoid overmixing.​

Bake the brownie base
Spread the batter evenly into the prepared pan, smoothing the top with a spatula. Bake for 24–28 minutes, or until the top looks set and a toothpick inserted near the center comes out with moist crumbs but no wet batter.​

Add the marshmallows
Remove the brownies from the oven and immediately sprinkle the mini marshmallows in an even layer over the hot surface. Return the pan to the oven for 2–3 minutes, just until the marshmallows puff and start to lightly brown. Watch closely—marshmallows can burn quickly.​

Make the chocolate frosting
While the marshmallows are puffing or right after, prepare the frosting. In a medium microwave-safe bowl or small saucepan, melt the 1/4 cup butter. Whisk in the 1/4 cup cocoa powder until smooth and no lumps remain. Add 1 3/4 cups powdered sugar and 3 tablespoons milk, then whisk until smooth, thick, and pourable. If needed, add a splash more milk or a little more powdered sugar to reach a frosting that will spread easily but not run off the brownies. Stir in vanilla if using.​

Frost the marshmallow layer
Take the pan out of the oven and, while both the marshmallows and frosting are still warm, immediately pour the frosting over the marshmallow layer. Use an offset spatula or the back of a spoon to gently spread it to the edges, coating all of the marshmallows. It’s okay if some marshmallows peek through; that gives a nice “Mississippi mud” look.​

Cool and slice
Let the brownies cool completely at room temperature until the frosting is set, 1–2 hours. For extra neat slices, you can chill the pan briefly in the fridge before cutting. Lift the brownies out using the parchment, then slice into 16 squares.​