Place a saucepan filled with 1 inch of water over medium heat and bring to a simmer. Set a heatproof bowl over the simmering water, taking care that the bottom of the bowl is not touching the water, and melt the chocolate and butter together, stirring occasionally. Once melted remove from the heat and let cool slightly.

In a small bowl, whisk together the rye flour, baking powder, and salt and set aside.

Place the eggs in the bowl of a stand mixer fitted with a whisk attachment. Whip on medium-high speed, adding the sugar a little bit at a time, until all the sugar is incorporated. Turn the mixer to high and whip until the eggs have nearly tripled in volume, about 6 minutes.

Reduce the mixer speed to low and add the melted chocolate-butter mixture and the vanilla. Mix to combine, scraping down the sides of the bowl as needed, then add in the flour mixture just until combined. At this point the dough will be very soft and loose, which is normal; it will firm up as it chills.

Refrigerate dough until it just firm to the touch, about 30 minutes. (The longer you chill the dough the harder it is to scoop.)

Preheat the oven to 350 degrees. Line two baking sheets with parchment paper. Remove the dough from the fridge and scoop with a rounded tablespoon onto the baking sheets, shaping the balls of dough into rounds and spacing them 2 inches apart. Top each mound of dough with a few flakes of sea salt, pressing gently so it adheres.

Bake for 8 to 10 minutes, until the cookies have completely puffed up and have a smooth bottom and rounded top. Remove the baking sheets from the oven and let cool slightly (the cookies may flatten a bit), then transfer to a wire rack and let cool completely. The cookies will keep up to 3 days in an airtight container.