Put pistachios, apricots, and sugar into a food processor bowl; process until mixture is finely chopped, pulsing as needed to achieve the texture of small, dried lentils.
Meanwhile, while you make the filling in the food processor, remove the box of phyllo sheets from refrigeration, letting them sit on the counter to remove some of the chill. (This will make them easier much easier to work with.)
Assemble the Baklava
Unfold one pack of the phyllo sheets and stack them so they lie flat on a rimmed baking sheet; cover the stack with plastic wrap, making sure there is excess plastic wrap covering all four edges of phyllo stack.
Dampen and wring out a kitchen towel and drape it on top of the plastic wrap; this will hold the plastic wrap in place and also prevents the phyllo from drying out. **Please Note: This damp towel part is a very key to working with the phyllo sheet! Just be sure the damp towel never actually comes into contact with the phyllo sheets themselves - it should always rest on the plastic wrap.
Melt the butter in a small saucepan over low heat; using a pastry brush, brush the bottom of a 9 x 13 baking pan (glass or metal is fine) liberally with some of the butter.
Repeat until you have layered and buttered 10 sheets, the sprinkle with 1/3 of the prepared filling, spreading out evenly. (Take care not to soak each sheet with too much butter since you'll eventually wind up with 40 sheets of buttered phyllo when you're all finished.)
Repeat layering and buttering with 10 more phyllo sheets, then sprinkle on another 1/3 of the prepared filling. Again, repeat layering and buttering with 10 more phyllo sheets, sprinkling on the final 1/3 of the prepared filling.
Top final layer of filling with the last 10 sheets of the buttered phyllo. Cover pan loosely with plastic wrap and place baklava into the freezer for 30 minutes - this will make cutting the baklava much easier.
Preheat the oven to 350 degrees F. Position the oven rack in the center of the oven.
Before baking the baklava, use a thin, sharp knife in a gentle sawing motion to cut the baklava on the diagonal at 1 1/2-inch intervals in a diamond pattern. Be sure to cut the pastry all the way to the bottom of the pan (an electric knife really works wells here).
Take care not to compress the pastry by pressing down on it too much; these incisions cut into the pastry mark the serving portions as well as providing pathways for the flavored syrup to penetrate the pastry after baking.
If there is any melted butter remaining, pour it over the baklava now. Bake the pan of baklava, uncovered, until golden brown, approx. 35-40 minutes.
Transfer the pan of baklava to a cooling rack and let cool completely. Run a knife along the cut lines of the baklava and around the edges to help the syrup absorb more evenly.
Remove the saucepan from the heat and stir in the cardamom and salt. Pour the syrup evenly over the entire surface of the baklava, allowing it to run down into the cut marks and along the sides of the pan. Let baklava to cool to room temperature before serving.
1. When making baklava, you need about 10 hands.....or at least FOUR. Make baklava when you have a partner available!
2. Baklava is best about 24 hours after the syrup is added. Don't make the syrup ahead of time - if it simmers too long, it will be the wrong consistency when poured over the baklava and will crystallize, keeping the syrup from running down into the incisions of the dough and around the edge of the pan.
3. Baklava will keep at room temperature for up to 5 days, though the texture changes over that time.