In a large skillet, melt butter over medium heat. Add yellow onion, celery, bell pepper, garlic, Cajun seasoning, salt, and black pepper; cook until softened and fragrant, 3 to 5 minutes. Stir in crawfish and flour until smooth and combined, about 2 minutes. Remove from heat, and stir in cheese and green onion.
Line a rimmed baking sheet with parchment paper.
In a medium Dutch oven, pour oil to a depth of 1½ inches, and heat over medium-high heat until a deep-fry thermometer registers 350°.
On a lightly floured surface, roll 1 pastry sheet into a 13-inch square. Using a 3-inch round cutter, cut 16 circles. Transfer to prepared pan. Repeat with remaining pastry sheet.
In a small bowl, whisk together egg and 1 teaspoon water. Brush egg wash around edges of half of dough circles. Place 2 tablespoons crawfish mixture in center of egg-washed dough circles. Top with remaining dough circles; using a fork, crimp edges to seal.
Fry pistolettes, 2 to 3 at a time, until golden brown, about 1½ minutes per side. Let drain on paper towels. Let cool for 5 minutes before serving or refrigerating in an airtight container up to overnight.
Recipe by Louisiana Cookin' at https://louisianacookin.com/crawfish-pistolettes/