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Bring 2 cups of the milk and 3 tablespoons of the sugar to a boil in a medium (I used a 3 quart) heavy-bottomed saucepan.
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While milk is heating, put the cocoa, cornstarch and salt into a food processor and whir to blend. Turn them out onto a piece of wax paper, put the remaining 4 tablespoons sugar, the egg and egg yolks into the food processor and blend for 1 minute. Scrape down the sides of the bowl, add the remaining 1/4 cup milk and pulse just to mix, then add the dry ingredients and pulse a few times to blend.
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With the machine running, very slowly pour in the hot milk mixture. Process for a few seconds, then put everything back into the saucepan. Whisk without stopping over medium heat–making sure to get into the edges of the pan–until the pudding thickens and a couple of bubbles burble up to the surface and pop (about 2 minutes). You want the pudding to thicken, but you don’t want it to boil, so lower the heat if necessary.
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Scrape the pudding back into the processor (if there’s a scorched spot, avoid it as you scrape) and pules a couple of times. Add the chocolate, butter, and peppermint extract and pulse until everything is evenly blended.
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Pour the pudding into a bowl and press a piece of plastic wrap or waxed paper directly onto the surface of the pudding to prevent a skin forming. Chill until ready to assemble tart.