Preheat the oven to 350°F. Butter an 8 x 8-inch baking pan, then line it with foil, leaving a 2-inch overhang on two sides. Butter the foil as well.
In a medium bowl, mix the melted butter and cookie crumbs together. Press the mixture evenly into the prepared pan. Bake for 10 minutes. The crust will appear very soft and not fully baked, but it will firm up after cooling completely, about 15 minutes.
In a mixing bowl, beat the cream cheese and sugar on medium-high speed until smooth and creamy, about 1 minute. Add the egg, sour cream, and peppermint extract, and beat again until combined, about 1 minute.
Pour the cheesecake mixture over the cooled crust and spread evenly using an offset spatula.
Bake for approximately 30 minutes, until the edges are set but the center still has a slight jiggle. Transfer to a wire rack and cool completely for about 1 hour. Refrigerate for at least 2 hours or overnight.
To serve, carefully run a knife around the edges of the foil, then lift the bars out using the foil overhangs. Remove the foil, cut into bars, and sprinkle with crushed candy canes.