Preheat the oven to 400˚ F. Line 2 baking sheets with parchment paper.
On a lightly floured surface, roll out the puff pastry into a 12-by-14-inch rectangle. With a short side facing you, cut the dough in half horizontally: To do this, 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 around the edges. Sprinkle 1 cup 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-wide strips (about 24 strips).
Twist each strip 4 or 5 times, then pinch the ends together to prevent unraveling while baking. Place the twists on the baking sheets. Beat the egg with 1 teaspoon water in a small bowl and brush onto the twists. Sprinkle with the remaining 2 tablespoons cheese.
Bake the cheese straws until golden brown, 12 to 15 minutes. Let cool slightly before releasing from the parchment with an offset spatula. Let cool completely. Serve immediately or store in an airtight container for up to 1 day.