Preheat oven to 425 degrees with rack in center position.
Roll out puff pastry to 12-inch square. Brush maple syrup over entire surface.
Cut dough into thirds in each direction and then cut each square in half diagonally, ending with 18 triangles. Place a breakfast sausage link on the lower third of each triangle and roll up, making sure to place each tip-side down on a parchment-lined baking sheet.
Whisk egg and 1 teaspoon water in a small bowl until lightly beaten. Brush each puff with egg wash and bake for 16 to 18 minutes until golden brown.