Melt bittersweet chocolate and margarine together in the top of a double boiler over barely simmering water, stirring frequently and scraping down the sides with a rubber spatula, about 5 minutes. Remove from heat and cool.
Combine potato starch and baking powder in a bowl.
Beat eggs, sugar, and vanilla extract together in a separate large bowl with an electric mixer on medium speed until light and fluffy, about 5 minutes.
Fold the cooled chocolate mixture, potato starch mixture, chocolate chips, and walnuts into egg mixture until just combined.
Drop spoonfuls of batter onto the prepared baking sheet.
Bake cookies in the preheated oven for 4 minutes, turn baking sheets the opposite way in the oven, and bake until cookies are crisp on the edges, about 6 more minutes.
Let cookies cool on sheets for about 5 minutes; transfer to wire racks to finish cooling.
Kosher for Passover baking soda and baking powder are available in specialty markets and online.