Build the base: In a large cast iron skillet, sauté the onions, mushrooms, and garlic in butter over medium-high heat until softened.
Add vegetables: Stir in the green beans and corn and cook until warmed through.
Make the roux: Sprinkle in the flour and cook, stirring constantly, for 3–4 minutes to eliminate any raw flour taste.
Create the sauce: Slowly pour in the broth, stirring to keep the sauce smooth. Stir in the heavy cream and cook until slightly thickened and heated through.
Add chicken: Stir in the chicken until combined, then remove the skillet from the heat.
Top with pastry: Cut the puff pastry into 1-inch squares. Starting around the outer edge of the skillet, arrange the pastry on top of the filling, slightly overlapping as you work toward the center.
Finish: Whisk the egg with a splash of water and brush over the puff pastry. Season with coarse salt and fresh ground pepper.
Bake: Bake at 400°F until the puff pastry is deeply golden and crisp, about 35–45 minutes.