In a microwave-safe bowl, melt the dark chocolate and mint chocolates in 30-second intervals until nearly melted. Whisk until smooth; set aside to cool.
In a medium bowl, whisk together the flour, cocoa and salt.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter until creamy. Add the granulated sugar and brown sugar and beat at medium-high speed until light and fluffy, about 3 minutes. Beat in the egg yolks and vanilla. Pour the chocolate into the mixer and beat just until incorporated. Add the dry ingredients and beat at low speed, scraping the side of the bowl occasionally, until smooth. Transfer the dough to a sheet of plastic wrap and pat it into a 8-inch disk; wrap it up and refrigerate until chilled and firm, at least 30 minutes..
Preheat the oven to 350°F. Line 2 cookie sheets with parchment paper.
Place Turbinado sugar in a shallow bowl. Roll a heaping tablespoon of dough into a 1½-inch ball, roll the ball in the sugar and place it on the prepared baking sheet. Repeat with the remaining dough, spacing the balls about 2 inches apart. Using your thumb, dowel or a melon baller, make an indentation in the center of each cookie. Bake for 10 minutes, or until slightly firm. Remove the cookie sheets from the oven. Using your thumb, dowel or a melon baller, press into the cookies again. Return the cookies to the oven and bake 4 to 5 minutes longer. Cool cookies for 5 minutes on baking sheet, then transfer to wire rack to cool completely.
To make the White Chocolate Filing, Put the white chocolate in a heatproof bowl. Put the cream into a microwave-safe bowl and microwave at high power until boiling, about 30 seconds. Pour the hot cream over the white chocolate and let stand until melted, about 30 seconds, then whisk until smooth. Stir in the peppermint extract. Fill the thumbprints with the white-chocolate ganache and refrigerate just until set, about 30 minutes.