Position an oven rack in the lower third of the oven. Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.

In a medium bowl, knead together the cookie dough and 1/4 cup flour until smooth. Lightly flour a work surface. Roll out the dough to 1/4-inch thick. Using a 2 1/2-inch cookie cutter, cut out circles from the dough. Knead together any scraps of dough and roll out again. Continue to cut out pastry circles until there are 24 pieces in total. Place the dough circles on the prepared baking sheets. Bake until the cookies are slightly golden around the edges, about 10 minutes. Cool for 10 minutes and transfer to a wire rack to cool completely. Reserve 1 parchment paper-lined baking sheet.

Combine 1 cup of chocolate chips and the cream in a small bowl. Place the bowl over a pan of simmering water and stir until the chocolate has melted and the mixture is smooth. Whisk in the peppermint extract and refrigerate the mixture for 1 hour. Using a 1/2-ounce cookie scoop or a tablespoon measure, place the chocolate mixture in the center of the flat side of 12 of the cookies. Place the remaining cookies on top and gently squeeze to distribute the filling evenly. Place on a baking sheet and freeze until the filling has set, about 25 minutes.

Combine the remaining chocolate chips and the vegetable oil in a small bowl. Place the bowl over a pan of simmering water and stir until the chocolate has melted and the mixture is smooth. Dip the top of each cookie in the melted chocolate and return to the baking sheet. Sprinkle the tops with crushed candy canes or peppermint candies. Refrigerate until firm, about 1 hour. Store in an airtight plastic container.