On a piece of parchment paper, measure out a 9 x 12 inch rectangle. Turn the paper over (so you won’t get ink/pencil on your bark), and place on a sheet pan.
Place the bittersweet chocolate in a heatproof bowl, and set it over a saucepan of barely simmering water (remember not to let the bottom of the bowl to touch the water). Stir the chocolate every once in awhile, until it is melted and very smooth (about 110° F). Remove from the heat, and pour the chocolate onto the prepared rectangle. Using a cake spatula or a knife, spread the chocolate to fill the rectangle. Sprinkle with half the crushed peppermints, and then chill until set, about 15 minutes.
While it sets, combine the semi-sweet chocolate, cream, and peppermint extract. Again, warm over the barely simmering water, stirring occasionally, until the mixture is smooth. Cool the chocolate until lukewarm, about 5 minutes. Pour the semi-sweet chocolate mixture over the first chocolate rectangle, and spread the semi-sweet chocolate in an even layer. Chill until very cold and firm, about 40 minutes.
Warm the the white chocolate over barely simmering water until melted and smooth. Quickly pour the white chocolate over the firm semi-sweet layer, and use a clean spatula to spread it over evenly. Sprinkle with the remaining crushed peppermints, then chill just until firm, about 20 minutes.
Move the parchment to a cutting board. Trim away any uneven edges of the rectangle, and then cut the bark crosswise into 2-inch-wide strips. Cut each strip crosswise into 3 sections, and then cut each section diagonally into 2 triangles.
Put the bark in an airtight container, with parchment paper between layers of bark to prevent them from sticking to one another. Store the chocolate bark in the refrigerator. Let stand at room temperature for 10 minutes before serving. The bark should last 2 or more weeks.