In the bowl of a stand mixer, cream the butter until fluffy, about 1 minute. Slowly beat in the oil, sugars, and extracts until combined. Beat in the eggs, one at a time, mixing well after each addition. Gradually add in the dry ingredients and mix on low-speed until incorporated, scraping the sides of the bowl as needed. Beat in the peppermint chips until combined. Cover loosely with foil or plastic wrap and chill for 2 hours. Chilling is MANDATORY.

Preheat oven to 350 degrees F. Line 2 baking sheets with silicone liners or parchment paper. Drop rounded Tablespoonfuls of dough onto the baking sheets about 1-2" apart. Bake for approximately 9-11 minutes, rotating pans halfway through the baking time to ensure even cooking. Cool completely on the baking sheets.

Once cookies are cooled, melt the white chocolate chips and heavy cream together in a small bowl for 30 seconds in the microwave. Stir, then heat for another 10 seconds until melted and smooth. Drizzle evenly over the cookies and immediately top with crushed candy cane sprinkles.