In a microwave-safe bowl, melt the bittersweet and mint chocolates at 30 second intervals until nearly melted. Whisk until smooth, then let cool.
In a medium bowl, whisk the flour with with the cocoa and salt.
In the bowl of a standing electric mixer fitted with the paddle, beat the butter until creamy. Add the granulated sugar and brown sugar and beat at medium-high speed unti light and fluffy, about 3 minutes. Beat in egg yolks and vanilla. Scrape the chocolate into the mixer and beat just until incorporated. Add the dry ingredients and beat on low speed, scraping down the sides of the bowl occasionally, until smooth. Transfer the dough to a sheet of plastic wrap and pat it into a 7" disc. Wrap and refrigerate until well-chilled, at least one hour.
Preheat oven to 350 and position racks on upper and lower thirds of oven.
Line 2 cookie sheets with parchment paper. Spread the Turbinado sugar in a shallow dish. Scoop up tbsps. of dough and roll into balls, then roll in the sugar and place on cookie sheets.
Use your thumb to make an indentation in the middle of each cookie.
Bake for 10 minutes, until slightly firm. Remove from oven. Using a melon-baller, press center of cookies again. Return to oven, shifting the cookie sheets and bake for 5 more minutes, just until dry but not hard. Transfer cookie sheets to racks to cool completely.
Put the white chocolate in a heatproof cup. Put the cream into a microwave-safe bowl and microwave on high power until boiling, about 30 seconds. Pour the hot cream over the white chocolate and let stand until melted, then whisk until smooth. Stir in peppermint extract. Fill center of cooled cookies with white chocolate mmixture and refrigerate until set, about 30 minutes. Store in airtight container for up to 1 week.