Preheat oven to 350°F. Line the bottom and side of an 8 inch by 8 inch baking pan with parchment paper. Alternatively, grease and flour the pan.
In an electric mixer with either the paddle or whisk attachment, beat eggs on a medium speed. Once the eggs start to get foamy, increase the mixer to high and add the sugar (just the white sugar) slowly, a little bit at a time. Once all the sugar has been added, allow the eggs and sugar to continue beating for about 10 minutes until it is thick, pale yellow, and forms a ribbon when dropped on itself.
Place the marshmallows in a 2 quart saucepan over low heat. As they start to melt, stir them. They will clump into a sticky ball. Add the butter and chocolate. Stir occasionally until all ingredients are fully melted. (Don’t sweat it if there are few unmelted marshmallows.) Remove from heat. Stir in the brown sugar and vanilla. Set aside for a couple of minutes to make sure the mixture cools completely.
With a spatula, gently fold in the beaten eggs to the chocolate mixture. Fold (do not mix) the batter until fully incorporated.
Gently fold in the flour and salt just until all the flour is fully absorbed.
Pour batter into prepared pan. Bake for 23-25 minutes, just until a toothpick inserted comes out clean. Cool completely in pan on wire rack.