Using a blend of greens creates a slightly bitter green filling with sweet notes from the sautéed aliums. Lean on store-bought dough from your grocer’s deli to crank out these savory hand pies quickly.
Place a pizza stone on bottom oven rack; preheat oven to 500°F. Working in batches, pulse greens, onion, and celery leaves in a food processor until finely chopped, about 5 times per batch.
Heat oil in a large skillet over medium-high. Add chopped greens mixture, leeks, salt, and pepper; cook, stirring often, until greens are very tender and liquid has evaporated, about 10 minutes. Remove from heat; let cool 20 minutes. Stir in mint and parsley.
Roll and stretch pizza dough into a 25- x 16-inch rectangle on a floured work surface. Cut out 8 (6-inch) rounds from dough; place 3 dough rounds on each of 2 baking sheets lined with parchment paper; set remaining 2 dough rounds aside. Roll the dough rectangle scraps into a 14-inch square. Cut out 4 (6-inch) rounds. Discard remaining dough scraps.
Divide cooled greens mixture evenly among the 6 dough rounds on baking sheets (about 1/2 cup per round). Lightly brush edges of filled rounds with water. Place remaining 6 dough rounds on top of each filled round; pinch edges to seal. Using your hands, flatten and spread filling to edges of each sealed pie. Make a 1/2-inch hole in center of each pie using a knife to allow steam to escape.
Transfer pies onto hot pizza stone in preheated oven; bake until golden brown, about 10 minutes. Let cool 5 minutes; serve warm.