Preheat the oven to 350 degrees Fahrenheit. In a stand mixer, beat the room temperature butter and sugar together until blended. Add the eggs, one at a time, just until blended, being careful not to overmix.
Melt the baking chocolate in the microwave at 50% power. Stir every 30 seconds until melted. Add the melted chocolate to the butter-sugar mixture and mix just until blended.
Add the flour and mix on low until blended. Using a wooden spoon or silicone spatula, stir in the vanilla extract and chocolate chips.
Spread the brownie batter into a greased 13 inch x 9 inch pan and bake at 350 degrees Fahrenheit for 35-40 minutes or until you can poke a toothpick through the center and it comes out clean.
Take the brownies out of the oven. Dust a light sprinkling of corn starch over the brownies. Then spread 1 ½ cups of miniature marshmallows over the top. Place the brownies back in the oven for 10 minutes or until marshmallows have slightly browned on the top.
Pull the brownies out of the oven and let them cool completely before drizzling chocolate over them.
In a microwave, heat the melting chocolate until it is a liquid. Using a spoon, drizzle the melted chocolate over the Mississippi mud brownies in several directions. Allow the chocolate to cool completely before cutting the brownies.
Enjoy your Mississippi mud brownies at room temperature or warmed up!