Preheat oven to 375F.
Roll the pastry out on a lightly floured surface to about a 12- by 14-inch rectangle. Make 6 cuts across the pastry at equal intervals the long way, then make 4 cuts across the pastry the short way (you end up with 24 pieces of dough).
Press each piece into a mini muffin cup. Cover the muffin tray with plastic wrap and transfer to the fridge to chill while you make the filling.
Whisk the eggs and egg yolk together, then measure out 1 tablespoon to reserve for the eggwash.
Heat the oil in a large lidded skillet over medium heat. Add the onion and cook until tender and starting to caramelize, about 5 minutes. Turn the heat up to medium-high, add the kale and water, cover the skillet, and cook until the liquid is evaporated and the kale is almost tender, about 8 minutes. (If the skillet gets too dry before the kale is almost tender, add a splash more water as needed.)
Stir in the garlic, dill, crushed red pepper flakes, salt, and pepper, and cook (uncovered) until very tender, about 10 minutes, stirring occasionally. Stir in lemon juice and turn off heat.
Cool slightly, then stir in the eggs (except the reserved egg for eggwash) along with the feta and pine nuts.
Fill the pastries with the kale mixture, using your finger to pack it down lightly. Lightly brush the dough with eggwash and bake until the pastries are puffed and golden, about 24 minutes.
Serve hot, warm, or room temperature along with chunky tomato sauce, if desired.