Preheat oven to 350 degrees. Line a 9 x 9 inch cake pan with parchment, leaving some overhanging the sides.
In a microwave-safe bowl, melt the butter and 1 cup of milk chocolate chips for 1 minute. Stir until the bowl is no longer warm. If needed, microwave an additional 20-30 seconds until completely smooth. Whisk in the sugar and brown sugar until combined.
In the bowl of a mixer, vigorously beat the eggs until very frothy and bubbly, about 3-4 minutes. With the mixer running, stream in the chocolate and butter mixture, then the espresso powder (if using) and vanilla. Mix in the flour, cocoa powder, baking powder, and salt until just combined. Folk in the remaining 1/2 cup of milk chocolate chips.
Spread the batter into the prepared cake pan. Bake for 35-40 minutes until an inserted toothpick emerges clean. Place the pan on a wire rack and cool for 30 minutes.
After 30 minutes, preheat the oven to 450 degrees.
Make the ganache. Melt the chocolate chips in a microwave-safe bowl for one minute, until completely smooth. If needed, heat an additional 20-30 seconds. Whisk in the heavy cream and vanilla.
Spread half of the ganache onto the brownies. Sprinkle the marshmallows overtop. Toast for 5-8 minutes until the mallows are golden brown to your liking. Cool the brownies at least 30 minutes, then drizzle the remaining ganache overtop. Slice and serve!
Notes
*For faster cooling: Set the tray of brownies in the freezer for 15 minutes.