Preheat the oven to 375 degrees F.
In a large bowl or pan, mix together the flour, sugar and baking powder. Add the shortening and work it in with your hands until well mixed. Add the eggs and vanilla, and 3/4 cup of water, and work in well. The dough should hold together but not be sticky. If needed, work in a little more water, 1 teaspoon at a time.
Place the seeds in a shallow baking pan.
Pinch off a small amount of dough, about the size of a small fist, and cover the remaining dough with a damp dish towel. Roll the dough out into a long rope, about 1-inch in diameter. Cut the dough into 2-inch pieces.
Drip a small amount of water through a fine mesh sieve, and sprinkle over a small portion of the sesame seeds to wet them (but do not wet the seeds until they will be used). Roll the dough in the seeds, pressing slightly to make them adhere. Place the seeded dough onto a large ungreased baking sheet. Repeat with the remaining dough.
Bake on the bottom rack of the oven until the bottoms are golden brown, about 10 minutes. Turn the cakes over and place the baking sheet on the top rack of the oven to bake for an additional 10 minutes.
Remove from the oven and cool on cooling racks. Serve at room temperature, or keep in tins for up to 2 weeks.
Yield: 40 cookies