In a small bowl, combine flour, baking powder, and salt. Set aside. Put chocolate and butter in a medium metal bowl and set over a saucepan of 1 in. simmering water (bowl should not touch water). Heat, stirring occasionally, until chocolate melts and is completely smooth. Remove bowl from pan and let cool.
Using a stand mixer with whisk attachment, beat eggs on medium-high speed until foamy. With motor running, gradually add sugar; then continue beating until mixture is light and tripled in volume, about 7 minutes. Add cooled chocolate and vanilla and beat to blend, scraping bowl as needed. Reduce speed to low and add dry ingredients, mixing until smooth (dough will be thin, like cake batter). Chill, covered, until dough is firm enough to scoop, at least 4 hours or overnight.
Preheat oven to 350° and line 2 rimmed baking sheets with parchment paper. Using a lightly oiled 1 tbsp. Measuring spoon, scoop a half-dozen portions of dough onto a work surface; chill remaining dough. Using lightly oiled fingers and the palm of your hand, roll 1 portion into a ball, then flatten into a disk about 2 in. wide. Repeat to make a second disk. Set a peppermint patty on one disk, top with the second, and press edges to seal. Repeat to make more cookies, spacing them about 11/2 in. apart on lined baking sheets.
Bake 1 sheet of cookies at a time (for the most even cooking) until set at edges and crackled in center, about 12 minutes. Let cool on baking sheets for a few minutes, then loosen from parchment with a wide spatula and transfer to wire racks to cool completely.
MAKE AHEAD Dough, up to 1 day, chilled airtight; or 1 month, frozen. Cookies, up to 2 days, airtight; or 1 month, frozen.
*Light brown, partially refined Demerara and turbinado sugars have a more complex flavor than regular brown sugar; find them with the baking supplies at well-stocked grocery stores.