In a large microwave-safe bowl, melt butter and chocolate in microwave at HIGH power for 40 seconds. Mix well. Whisk 2/3 cup granulated sugar, 3 Tbsp cocoa, vanilla, salt, milk, and egg yolk into chocolate mixture. Add flour mixture. Stir until thoroughly mixed. Pour batter into slow cooker. Spread evenly.
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Whisk together remaining granulated sugar, remaining cocoa, brown sugar, and hot water until sugar is dissolved. Pour over batter in slow cooker. Do not stir. Cover and cook on high 2 hours. The cake will be very moist and floating on a layer of molten chocolate when it is done. You'll know it's done cooking when its edges begin to pull away from the sides of the insert and nearly all of the cake is set.
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