Make the cookie dough. Melt chocolate gently in a heatproof bowl set over a saucepan of simmering water or in the microwave with short bursts and frequent stirring. Set aside to cool slightly.
Sift flour, cocoa powder, baking soda and salt into a medium bowl and whisk to blend evenly.
In a large bowl, cream together butter and sugar using a wooden spoon or rubber spatula until light and fluffy. Mix in yolk and vanilla extract. Stir through melted chocolate. Add flour mixture and fold it in until just incorporated.
Form the dough into an 11 to 12-inch log and wrap in plastic wrap. Refrigerate until firm, at least 30 minutes.
Preheat your oven to 350°F and line two large baking sheets with parchment paper.
Slice the chilled log into 16 equal portions. Roll each portion of dough into smooth balls and place onto prepared baking sheets spacing them about 2 inches apart. Use your thumb or a dowel (such as the back end of a wooden spoon) or a ¼ teaspoon measuring spoon to press down on the dough balls to create an indent. I prefer the back of a wooden spoon to create a deep well. Once you’ve made the impression with the dowel, move it around inside the well to open up the space to make room for lots of filling. If the dough cracks, press and patch it back with your fingers.
Place the baking sheet in the fridge and chill the prepared indented dough balls for 20 minutes. You can chill all the dough together on one baking sheet and then separate it out on multiple baking sheets when ready to bake). Bake for 10-12 minutes, then remove from the oven and transfer trays to a wire rack and lightly press down on the centers again with a ¼ teaspoon to reinforce the indents again as they may have lost their indent a little as they baked.
Let cool completely on trays as these cookies are delicate when warm.
Make the filling. Combine chopped white chocolate, cream and peppermint extract in a small saucepan over very low heat and stir until melted and smooth. Spoon the filling into the wells of the cooled cookies, sprinkle crushed peppermint over top and refrigerate until set.