Place 2 dozen small candy cups on a sheet pan or in a mini muffin tin.
Heat double boiler or sauce pan with water until gently simmering. Place a large heatproof bowl over water, and place all but 3/4 cup dark chocolate in bowl, and stir until one-third of the chocolate is melted.
Remove the bowl from the sauce pan, and stir in remaining chocolate, just until melted and smooth. Wipe off bottom of bowl so that no water comes in contact with the chocolate. Stir in 3/4 teaspoon peppermint extract.
Using a small ladle or spoon, fill 24 candy cups with the chocolate, a little less than 1/2 way full. Set aside. Do not refrigerate.
While the chocolate is setting up, repeat steps 2 and 3 using the same procedure with the white chocolate, using the remaining 3/4 teaspoon peppermint extract.
Pour, spoon or ladle white chocolate into cups, coming all the way to the top of the cups.
Sprinkle with candy cane dust, chunks and chips, pressing them gently into the white chocolate.
Set aside at room temperature until firm, about 1 hour.
Store in an airtight container for up to two weeks.