In a bowl, combine the flour, cocoa powder, sugar, baking powder and salt.
Add the 3 beaten eggs, melted butter, and vanilla extract, and stir everything together into a soft, sticky dough. Let cool for 10 minutes, then stir in the mini chocolate chips.
Cover the bowl and chill the dough in the refrigerator for 3-4 hours, until well chilled.
Preheat the oven to 350 F, and line a large baking sheet (or two medium baking sheets) with parchment baking paper.
Divide the chilled dough into three portions (about 10 ounces each), dusting each portion with a little flour to keep the dough from sticking to your hands. Roll or press each portion of dough out into a rectangular-shaped log, measuring about 3x8 inches and about 3/4 inch thick.Make sure you leave about 4 inches between each portion of dough on the baking sheet, since they will spread out as they bake. Use two baking sheets if you need more space. Whisk up the last remaining egg, and lightly brush the top of the dough with the beaten egg.
Bake for 20-25 minutes, until the edges are starting to brown, and the biscotti has spread out and puffed up a bit. If you gently press on the center of one, it should feel baked through, not raw or doughy.
Remove from the oven, and let cool for exactly 10 minutes on the baking sheet.Reduce the oven temperature to 325 F. Use a large spatula or cake lifter to carefully lift the slabs of biscotti onto a cutting board. Use a sharp knife (I find that a chef's knife works best) to slice the slabs into individual biscotti about 3/4 - 1 inch thick. Gently place the biscotti back onto the baking sheet, turned on their side with the cut side facing up.
Bake the biscotti for 10 minutes. Turn the biscotti over, so the other cut side is facing up, and bake for 10 more minutes. Cool completely.
Make sure the biscotti is completely cooled and hardened before storing it in an airtight container.
For the melted chocolate, melt the 1 1/4 cups chocolate chips in a double boiler until smooth. You can use a spoon or piping bag to just drizzle the chocolate over the biscotti, or dip one end in and shake off the excess. I used a spoon to spread and swirl the melted chocolate along one side of the biscotti.Place on a parchment-lined baking sheet for 1-2 hours until the chocolate sets and hardens.