Preheat oven to 375 degrees. Line a large baking sheet with parchment; set aside.
Make your pastry: In a large bowl, whisk together the flour, salt and sugar. Using a pastry blender or two knives, cut in the cold butter and the shortening until the mixture resembles coarse meal.
In a small bowl or measuring cup, whisk together the buttermilk and egg. Pour over the flour mixture and, using a fork, stir to combine just until the dry ingredients are moistened. The mixture will be very crumbly.
Turn the dough onto a flour-lined surface. Knead the dough very gently - just a couple of times - to bring it together, then divide in half. Roll each half into a 13x9-inch rectangle - you can use a ruler to measure, or you can cheat (like I did) and use a 13x9-inch pan to check for size and cut around. Trim around the edges of the rectangle so that you have clean, straight edges (again, the pan comes in handy here), and cut each rectangle into thirds in both directions, giving you 9 equal-sized rectangles from each larger rectangle (18 total small rectangles).
In a small bowl, whisk together the egg yolk and water. Brush 9 of the pieces with the egg wash, then spoon on about 2 teaspoons of fig butter. Sprinkle the fig butter with a small amount of brown sugar - maybe 1/2 teaspoon for each piece. Top with one of the remaining dough pieces, pressing along the edges slightly. Using a fork, crimp along the edges of each tart and poke a few holes in the top to allow steam to escape.
Place the tarts on the prepared baking sheet. Brush the tops with some of the remaining egg wash, then sprinkle with a bit more brown sugar. Bake for about 25 minutes, until golden brown.
Store cooled pop tarts in a zip-top bag for up to 3 days. To reheat, toast in a toaster set to the medium setting.
Notes
I suspect that these would freeze quite well, though I have not tried it myself. To do so, freeze unbaked tarts on the lined baking sheet, then transfer to a zip-top bag. Bake straight from the freezer, adding extra time to the baking time.