Prepare cookie sheets by lining with parchment paper or spraying with cooking spray. Oven heats to 350 degrees but it's easiest to do this later as the dough needs to be in the freezer for 5-10 minutes.
In the bowl of an electric mixer, or use a handheld blender, beat the butter, granulated sugar, and brown sugar until smooth and combined. Add the egg, vanilla and peppermint extract. Beat until well combined and smooth.
In a separate bowl, whisk together the flour, dry pudding mix, and baking soda. Add the dry ingredients, 1 cup at a time, to the wet ingredients in the mixer until just combined. The dough will be thick and chunky.
Mix in the white chocolate chips on low speed just until incorporated.
Grab about 2 tablespoons of dough and roll into a ball (about the size of a golf ball) and flatten it in the palm of your hand. Place 1 unwrapped candy cane kiss in the middle, and then bring up the sides and form into a ball. Place the dough balls on a plate and put them in the freezer for about 5-10 minutes before baking. ** I roll and stuff 12 cookie dough balls and freeze those for about 10 minutes. While that batch is freezing, I will roll the other 12 dough balls and freeze those while the first batch is baking in the oven. This way, some dough balls are not in the freezer for 20 minutes or more.
Place dough balls onto the prepared cookie sheet and bake for 11-14 minutes. They will still be pale but should not look wet/raw dough. Let cool on cookie sheet for 30 minutes and then drizzle with melted candy cane kisses.
To make the drizzle : melt 1 cup of unwrapped candy cane kisses + 1/2 teaspoon canola or vegetable oil, in the microwave for 1 minute, stirring after 30 seconds. Drizzle this over the cookies (after they've been cooling for about 30 minutes). Remove cookies to cooling rack to cool completely (chocolate drizzle will harden).