The egg wash at the end is completely optional, but if you've gone through all the work already, why not give it that last little sparkle? Bon appetit!
3 tablespoons unsalted butter
1 small yellow onion, minced
1 stalk celery, thinly sliced
2 carrots, diced medium
5 tablespoons all-purpose flour
2 cups chicken broth
1/3 pound green beans, trimmed and cut into 1-inch pieces
2 cups diced cooked chicken
Salt and pepper
1 sheet thawed puff pastry
1 beaten egg
In a large saucepan, melt butter over medium-high. Add onion, celery, and carrots and cook until onion is translucent, about 4 minutes. Add flour and cook, stirring, 1 minute. Gradually add broth, stirring constantly, then bring to a boil. Reduce heat and simmer 5 minutes. Stir in green beans and chicken and season with salt and pepper.
Divide filling among four 8-ounce ramekins. Roll out 1 sheet puff pastry and cut 4 rounds to fit ramekins. Cut a small X in center of each to vent, then place on top of filling. Brush with beaten egg. Bake at 375 degrees until filling is bubbling and crust is golden, 30 to 40 minutes.