Melt vegetable shortening in the microwave. Remove and let cool slightly. In a large mixing bowl, sift together dry ingredients. Stir in finely chopped chocolate and minced rosemary. Set aside.
In a medium mixing bowl, whisk together eggs, sugar, and vegetable shortening until very well combined. Stir the wet ingredients into the dry ingredients with a sturdy spatula. Continue mixing until the mixture is very crumbly and all the liquid has been absorbed.
Pick the dough up in your hands and squish and knead it together until no dry bits of flour remain. It shouldn't need more liquid, but if it does you can add water ½ teaspoon at a time (sprinkled directly onto the dried bits) until it comes together.
Press the dough into a 1" thick disc. Wrap in saran wrap and refrigerate at least 1-2 hours until very firm to the touch.
Chocolate Halva Ganache
Place roughly chopped chocolate and tahini in a metal or glass heat-proof (not plastic or rubber bottomed) medium sized bowl that fits over the mouth of a small sauce pot without touching the bottom. Set aside.
Fill the pot with about an inch of water. Bring to a rolling boil, place the bowl of chocolate on top, then turn the heat off. The residual steam will be enough to melt the chocolate. Stir the chocolate and tahini together with a spatula until smooth and completely melted.
Place halva in the bowl of a small food processor. Pulse until it has the consistency of wet, clumpy sand. Add the melted chocolate and process until smooth. Taste and add sugar and salt if needed, processing again to combine.
Pour the chocolate halva ganache back into the bowl you melted the chocolate in. Stir in the sesame seeds. Cover with plastic wrap and set aside to cool at room temperature.
If not using immediately, refrigerate. Remove from the fridge 30 minutes before use to warm up slightly.
Hamantaschen Assembly
Preheat oven to 350°F and arrange a rack so it's in the center of the oven.
Lightly dust your counter and rolling pin with flour and roll the dough to about ¼" thick. If the dough is still slightly chilled it might be crumbly — knead it with your hands on the counter bit until it softens up.
Use a 3" round cookie cutter to cut as many dough rounds as you can. Knead any scraps briefly to recombine, then wrap in plastic and return to the fridge to chill slightly before re-rolling.
Scoop about 1 teaspoon chocolate halva ganache into the center of each cookie round.
Dip your finger in a bit of water and run it around the edge of the cookie. Then fold the sides of each circle into a triangular shape, pinching to firmly seal the edges shut. If there are gaps of air inside the cookie that's okay — the filling will melt and fill in as they bake.
Repeat until all the dough and filling has been used up. Arrange cookies in tight rows on a quarter sheet pan. Chill assembled cookies in the refrigerator (not freezer) for 30-60 minutes, no need to cover.
To bake, arrange chilled cookies about 2-3 inches apart on a lined sheet pan. Bake for 14-16 minutes until the corners of the cookies are beginning to brown just slightly.
Let the finished cookies cool on the pan for 5-10 minutes before transferring to a wire rack to cool completely.
RECIPE NOTES
Prep time includes assembly.
You may need to bake the cookies in batches or use multiple sheet pans to fit them all in the oven at once. If you use multiple racks in the oven, just swap the sheet pans halfway through baking so the cookies brown evenly.
If you're not Jewish and you feel weird making a Jewish holiday treat, you can shape these into squares by pinching four corners instead of three.