Preheat the oven to 350 degrees. Butter or spray a 9-inch pie plate or spring form pan. Set aside. Whisk together the flour, cocoa powder, baking powder and salt in a bowl.
In a medium bowl, mix the butter and sugar with an electric hand mixer, beat in the eggs and vanilla for 1-2 minutes, until the mixture becomes fluffy and light in color.
Add to the dry ingredients and mix until smooth, being careful not to over-mix as this will cause the brownie to be more cake-like in texture. Pour into the prepared pan.
Bake on a middle shelf of the oven for 25 to 30 minutes, or until a toothpick inserted in the center comes out clean.
While the brownie is baking make the chocolate frosting. Combine all ingredients together in a medium bowl and mix on medium low speed with a hand mixer, until smooth.
Remove from the oven and immediately sprinkle the marshmallows on top and return to the oven for a few minutes to melt the marshmallows.
Remove from the oven and carefully spread the chocolate frosting over top. Let cool completely. Cut into wedges to serve.