TO MAKE SPICED CHOCOLATE SHORTBREAD (PREPARE THIS THE NIGHT BEFORE): In a large bowl, cream unsalted butter and dark brown sugar together until light and fluffy, approximately 5 minutes. Then, beat in the molasses. Sift flour, ground ginger, ground cinnamon, ground allspice, ground white pepper, and salt directly into the bowl and mix until fully incorporated. Then add the chopped semisweet chocolate, and mix until even. Transfer the dough onto a large sheet of parchment paper. Press and roll it into a thick log, then wrap it tightly with the parchment. Chill in the fridge for 1 hour, or until firm. This can be made days ahead.
TO COOK AND BAKE THE LIDDED OATMEAL: Preheat the oven on 325° F (165° C).
Combine quick oats, whole milk, water, light brown sugar, and salt in a pot. Bring to a simmer, then cook for a few minutes until the oatmeal has fully thickened. Divide into individual serving cups, then set aside (you'll be able to fill 6 to 8 small cups).
(If your shortbread dough has gotten too hard to roll out, cutting it into small pieces and kneading it back together should help soften it.) Dust the dough with a bit of powdered sugar, then roll it out to 1/2-inch (12 millimeters) thickness. Use a cookie or biscuit cutter that's the same size as the cups you're using, cut out however many lids you need (you may not need all of the dough). Knead the scraps back together and roll out again if needed. The ratio of the cookie and oatmeal by volume should be 1 part cookie to 3 parts oatmeal.
Transfer the lid into the serving cups, right over the cooked oatmeals. Bake in the oven for 22 to 25 minutes, until the shortbread lids are golden brown and slightly puffed up. Dust with a little powdered sugar and serve while hot.