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Preheat the oven to 350 degrees F. Grease an 8-inch-square pan with butter, set to the side.
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In a medium mixing bowl, whisk together flour, baking powder, and salt. Set off to the side.
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In a small saucepan, bring two inches of water to a boil. Add butter and unsweetened chocolate to a medium mixing bowl. Place heat-proof mixing bowl over the simmering water. Stir with a rubber spatula until all the chocolate butter are melted and combined. Remove bowl and allow to cool slightly.
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Whisk the sugars into the melted chocolate and butter. Whisk in eggs and yolk, one at a time. Add flour all at once, fold together using a rubber spatula. Stir in chocolate chips when almost all incorporated.
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Pour batter into baking dish and bake for 25 to 30 minutes. While the brownies bake, prepare the marshmallow layer.
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Pour ¼ cup cold water into the bowl of an electric stand mixer, fitted with the whisk attachment. Pour the gelatin envelopes on top of the water and let sit.
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In a small saucepan, combine the other ½ cup cold water, sugar, corn syrup, and salt. Heat over medium/high heat. Stir gently until mixture starts to bubble. Heat until mixture reaches 240 degrees F.
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Turn the mixer on low speed and carefully drizzle in the hot sugar liquid. Once all the sugar is added, turn the mixer speed up to medium/high and whip mixture until white and thick, about 12 minutes.
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When the marshmallow has cooled slightly, add ½ cup crushed candy cane bits to the marshmallow and mix to combine. Pour the marshmallow over the slightly cooled brownies, top with remaining smashed candy cane bits, and chill for at least an hour or, for best results, chill overnight.
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Cut into squares and serve. Brownies will last at room temperature in an air-tight container for up to five days.