Preheat oven to 375 degrees F. Line baking sheets with parchment paper.
Flour your work surface, so the puff pastry does not stick. Roll out your puff pastry until it measures 10 by 15 inches. Using a ruler, cut the rectangle into 2 (5 by 15-inch) rectangles. Then cut each rectangle into 5 (5 by 3-inch) rectangles. Repeat with the other sheet of pastry. Transfer the 20 rectangles onto your baking sheets and chill until they are cold.
Spread a small amount of spicy mustard onto the center of 10 of the rectangles. Add a small amount of chicken, broccoli, and shredded cheese onto each triangle. Be careful not to overfill.
Brush borders with egg wash and top with the remaining 10 rectangles. Use your fingers to gently press the edges together. Press the edge of a fork around border to completely seal pockets. Brush pockets with remaining egg wash and cut 3 - 4 (i-inch) slits into the top of each pocket.
Bake hot pockets, rotating once, until the tops are golden and the filling is bubbling, about 25-30 minutes. Cool on a wire rack before serving.
Recipe makes 10 homemade hot pockets.