Preheat the oven to 400 degrees F. Line two 18-by-13-inch sheet pans with parchment paper.
Dust a work surface lightly with flour. Roll out the puff pastry into a 14-by-12-inch rectangle. With the short side facing you, cut the dough in half horizontally: Measure 7 inches up the side, mark the dough and cut in half, using the mark as a guide.
Evenly spread the fig jam over the bottom half of the dough, leaving a little room at the edges. Sprinkle 1 cup of the cheese over the jam. Top with the other half of the dough, pressing down lightly to adhere to the cheese and jam. With a pizza cutter, cut into 1/2-inch strips (about 24 strips).
Twist each strip four or five times, then pinch the ends together to prevent unraveling while baking. Place the twists on the prepared baking sheets.
Mix the egg with 1 teaspoon water in a small bowl and brush onto the twists. Sprinkle with the remaining 2 tablespoons cheese.
Bake until golden brown, 12 to 15 minutes. Let cool slightly before releasing the straws from the parchment with an offset spatula. Let cool completely.
Serve immediately or store in an airtight container for up to 1 day.