Sift together the flour, nutmeg, cinnamon, allspice, baking powder, and salt and set aside. In the bowl of a standing mixer fitted with the paddle attachment, mix together the olive oil, vermouth, and eggs until well blended. Add the dry ingredients all at once and mix on low speed until the mixture forms a cohesive mass. Scrape the dough onto a very lightly floured board. Knead lightly until smooth. Shape the dough into a ball, flatten, and place in a bowl. Dust the top lightly with flour, cover with a towel or plastic wrap, and refrigerate for at least 2 hours or up to overnight to allow the dough to relax.
If necessary, lightly dust the work surface with flour. Cut the dough into 6 or 8 pieces. Roll each piece into a rope 1/2-inch in diameter. Cut the ropes into 1/2-inch pieces. Shape into "gnocchi" with a gnocchi board, butter paddle, or on the back side of the tines of a fork. The deeper the grooves on the cookies the better as they puff up as they fry and the ridges, unless deep, tend to disappear.
Heat the oil in a deep-fryer or deep pan to 350 degrees F.
Fry the cookies in about 4 batches until puffed and medium to dark brown, about 1 1/2 minutes. It is easiest to pile the cookies in a sieve or fry basket and then dip them into the hot oil. Each batch should include enough cookies to cover the surface of the oil. Any fewer and the oil gets too hot and browns the cookies before they are cooked through. Stir the cookies while they fry so they do not stick together. The number of cookies you add to the oil and the amount of oil will change the timing. The cookies first turn light gold, then a light brown, and finally almost a light milk chocolate color. Remove just as they get to this color or a little before. Drain on paper towels.
While still hot, put the cookies in a large shallow dish and pour the honey over them. Toss the cookies several times as they cool. If the cookies absorb all the honey, add more - they should be fully saturated. Serve in a pretty bowl or store in a covered container. They will keep for about 1 week.
Chef's Notes: Starting with a 1/2-inch piece of dough yields a cookie about 1 inch long, perfect for popping in your mouth all at once. They can be made smaller or larger, depending on your preference. Two hundred sounds like a lot of cookies. But once you taste one you will understand why you should be thinking about making several batches in quick succession.