Preheat the oven to 400°F.
Melt the butter in a large, heavy-bottomed skillet over medium-high heat. Add the onion, shallot, ginger, and garlic and cook, stirring, until translucent, but not brown, about 5-7 minutes. Sprinkle the flour on and stir to incorporate. Pour the chicken stock in a thin stream into the skillet, whisking constantly. Bring to a simmer to thicken.
Add the carrots, celery, and chicken. Simmer, stirring often, until the chicken is cooked through, about 10-15 minutes. While the chicken is simmering, trim the puff pastry into circles a little bit larger than the baking dishes you’ll be using.
When the chicken is cooked, stir in the green onions, taste and season with salt and pepper. Spoon equally into 4 mini baking dishes. Top the baking dishes with the puff pastry and use a fork to crimp and adhere. Brush the tops with the lightly beaten egg. Place in the oven and bake until tops are puff and golden brown, about 20-25 minutes. Enjoy warm!
