Thaw the puff pastry on the counter for 30 minutes. Test to be sure it's pliable before proceeding; if not, let it rest for another 10 minutes.
Preheat the oven to 400°F.
Pit the cherries and slice each in half.
Toss the cherries with the lemon juice and granulated sugar in a small bowl.
Place the puff pastry on a piece of parchment paper on a baking sheet. Using a rolling pin, gently roll it out until it's roughly a 12" square.
Use a pizza cutter or knife to slice the pastry into four equal squares.
Spoon four cherry halves in a line in the center of each pastry square.
Brush the beaten egg on the edges of the pastry, then lift one corner of the pastry and gently fold it over the cherries to meet the opposite corner. Use a fork to press the dough edges together and seal the cherries inside.
Brush the remaining egg mixture over the pastries, and sprinkle the tops with extra sugar, if desired.
Bake for 18 to 20 minutes, until light brown and puffy.
Let cool for 10 minutes before serving.
