In a medium size bowl whisk together well the flour, cocoa, baking powder, salt, cinnamon, nutmeg, and allspice. This takes longer than you may think; stir with the whisk for 1 full minute.
In another medium size bowl, beat the eggs with a hand-held electric mixer or a stand mixer with the whip attachment on medium to medium-high speed for 1 to 2 minutes, until the eggs are full of small bubbles. Continue beating, and gradually sprinkle in the granulated sugar. Beat until the eggs thicken slightly and become whitish in color, 2 to 3 minutes more.
Scrape the bowl. While beating on medium speed, slowly drizzle in the olive oil in a fine stream. This may take 2 minutes; it’s like making mayonnaise. On low speed, beat in the milk and vanilla.
With a wooden spoon, stir in the dry ingredients in two installments, mixing until smooth after each; then stir in the nuts and chocolate chips. The dough will be very thick and will hold its shape in the spoon and feel a bit tacky. Cover the bowl loosely with a kitchen towel and let the dough rest 30 minutes.
Meanwhile, adjust an oven rack to the center position and preheat the oven to 350 degrees F. Line two heavy 14- x 17-inch cookie sheets with cooking parchment or silicone baking pan liner. (If you only have one cookie sheet, cool it completely before baking the second batch of cookies).
To shape the cookies, roll rounded teaspoonfuls of dough between your palms into smooth shiny balls measuring about 1 inch. The oil in the dough will keep it from sticking to your hands. Space the balls about 2 inches apart on the prepared sheets, 18 cookies to a sheet.
Bake 1 sheet at a time, until the cookies smell fragrant and their tops feels dry and have numerous cracks, about 10 minutes. Do not overbake. The cookies should be tender, not dry. They will almost double in size during baking.
Cool the cookies on their baking sheets for 3 to 5 minutes, then transfer them to wire cooling racks to cool completely. Set the racks of cooled cookies over some baking parchment or wax paper.
To make the icing, in a small bowl use a hand-held electric mixer to beat the powdered sugar, butter, 2 tablespoons milk, and vanilla together until smooth and thick. The icing should have the consistency of heavy cream. If necessary, beat in more milk by droplets.
One at a time, dip the tops of 5 or 6 cookies in the icing and set them upright on the cooling rack. Dust them immediately with the colored sprinkles or jimmies. Repeat with the remaining cookies. You can reuse the sprinkles that fell onto the parchment or wax paper. Let stand until the icing has set, about 1 hour. Serve cookies or store them.
Storing. Store airtight at room temperature for 3 to 4 days. To freeze, place the cookies on a parchment-or foil-lined cookie sheet and freeze solid. Transfer the cookies to heavy-duty re-sealable plastic bags and freeze for up to 2 weeks. Thaw the cookies in the bags completely before removing and serving.