Preheat oven to 350 degrees F (175 degrees C). Grease 2 8x8-inch baking pans.
Melt unsweetened chocolate and butter in the top of a double boiler over simmering water; stir to mix well. Beat the eggs and egg whites together in a mixing bowl with an electric mixer on high speed until very fluffy, about 5 minutes. Gently stir the butter and chocolate mixture, sugar, coffee, and vanilla flavoring into the egg mixture, and beat well to combine.
Mix together all of the potato starch, all of the matzo cake meal, cocoa powder, and salt in a bowl. Beat the dry ingredients into the chocolate mixture with electric mixer until fluffy and thoroughly combined. Gently fold the chocolate chips into the batter, and divide between the 2 prepared baking dishes.
Bake in the preheated oven until a toothpick inserted into the center of a pan shows moist crumbs, 25 to 30 minutes.