Preheat oven to 350°F. Butter an 8-inch, square pan and line with parchment paper.
Melt half of butter and then mix with crumbs, brown sugar and a pinch of salt. Press firmly into bottom of pan. Bake in bottom third of oven until set, 10 to 12 minutes.
Combine remaining butter and unsweetened chocolate in a heatproof bowl; microwave at 20-second intervals until melted, stirring after each interval. Cool slightly. In a separate bowl, whisk sugar, vanilla and eggs. Stir in chocolate mixture. Add flour and mix until just combined. Pour over cooled crust and bake until just set, 22 minutes.
Remove from oven. Set broiler on high and place oven rack 3 inches from heat. Sprinkle marshmallows and chocolate chips over cookie and broil until marshmallows start to brown, 1 minute. Remove from oven and let cool before cutting into squares.