Step 1: Place a saucepan filled with 1 inch of water over medium heat and bring to a simmer.
Step 2: 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.
Step 3: Once melted remove from the heat and let cool slightly.
Step 4: In a small bowl, whisk together the rye flour, baking powder, and salt and set aside.
Step 5: Place the eggs in the bowl of a stand mixer fitted with a whisk attachment.
Step 6: Whip on medium-high speed, adding the sugar a little bit at a time, until all the sugar is incorporated.
Step 7: Turn the mixer to high and whip until the eggs have nearly tripled in volume, about 6 minutes.
Step 8: Reduce the mixer speed to low and add the melted chocolate-butter mixture and the vanilla.
Step 9: Mix to combine, scraping down the sides of the bowl as needed, then add in the flour mixture just until combined.
Step 10: At this point the dough will be very soft and loose, which is normal; it will firm up as it chills.
Step 11: Refrigerate dough until it just firm to the touch, about 30 minutes. (The longer you chill the dough the harder it is to scoop.)
Step 12: Preheat the oven to 350 degrees.
Step 13: Line two baking sheets with parchment paper.
Step 14: 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.
Step 15: Top each mound of dough with a few flakes of sea salt, pressing gently so it adheres.
Step 16: Bake for 8 to 10 minutes, until the cookies have completely puffed up and have a smooth bottom and rounded top.
Step 17: 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.
Step 18: The cookies will keep up to 3 days in an airtight container.
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