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In a blender or food processor, add the sugar and chocolate pieces and pulse to process until the chocolate is very finely chopped.
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In the microwave or on the stove, heat the whole milk until just bubbling around the edges.
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Add the hot milk to the chocolate and sugar in the blender and process until smooth.
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Pour chocolate mixture into a medium-sized bowl and let cool completely.
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Stir in the heavy cream and vanilla and chill for 30 minutes or longer.
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Add to your ice cream maker and follow the manufacturer's instructions. When transferring to an airtight container, break the brownies into bite-sized pieces and drop into the ice cream in layers. Freeze for several hours or overnight.