Preheat oven to 350 degrees. Line 2 large baking sheets with parchment paper and set pans aside.

Beat the butter, powdered sugar, vanilla, and salt in the bowl of a stand mixer fitted with the paddle attachment until mixture is smooth, about 2 minutes on medium speed. Stop mixer and scrape down the sides and bottom of the bowl. Turn mixer to low and gradually add the flour, increasing speed to medium and mixing until combined. (Mixture will be very crumbly.)

Place the granulated sugar in a small mixing bowl. Squeeze dough into balls of about 2 teaspoons of dough, then roll in the sugar. Arrange balls 1 inch apart on prepared baking sheets. Bake 10 minutes, then remove from the oven and press your thumb (or the handle of a wooden spoon) in the center to make deep indentations. Return cookies to the oven and bake until golden around edges, about 7 to 10 minutes more. If the indentations begin to disappear, remove from the oven and press your thumb in again.

Cool cookies completely on wire racks.

Combine the 6 tablespoons unsalted butter, white chocolate, and corn syrup in the top of a double boiler set over just-simmering water. Whisk occasionally until chocolate melts then whisk vigorously to make a cohesive mixture. Stir in the 2 tablespoons crushed candy canes, the 1/2 teaspoon peppermint extract, and food coloring (if using). Remove from the heat and cool slightly, until mixture starts to thicken.

Fill thumbprints with the chocolate mixture and sprinkle with additional crushed candy canes if desired. Allow chocolate to set, then serve. (Cookies can be stored in an airtight container at room temperature for up to 3 days, or in the freezer for several months.)