Preheat oven to 350 degrees. Butter the bottom and sides of a 9-inch tart pan and set aside.

To make the crust, add the flour, sugar and salt to the bowl of a food processor and pulse once to combine. Add the butter, then process until the mixture comes together and forms large clumps, about 30 seconds to one minute. Distribute the dough around the tart pan and press it evenly around the base and up the sides. Use the bottom of a glass to help even it out if needed. Bake for 15 minutes, until lightly golden. Remove from the oven and allow to cool before filling.

To make the filling, don't bother cleaning out the food processor and add the pistachios, sugar, flour and salt, and pulverize until they form a fine powder. Add the butter and process until evenly distributed, then add the egg and vanilla, and blend until smooth.

Pour the filling into the cooled crust (it's okay if it's still a little warm, just not blazing hot) and spread it out into an even layer. Take one sliced apricot half at a time and slide it on to a spatula and gently fan it out with your hands. Slide it off the spatula and onto the pistachio filling, then repeat with the remaining apricot halves to form a pretty pattern. Alternatively, you could leave the apricot halves unsliced and just place them on the tart as-is.

Bake for one hour, or until a toothpick inserted in the center of the pistachio filling comes out clean. Allow the tart to cool completely before serving. I personally prefer it served ice cold (and it slices much cleaner that way, too).