Position an oven rack in the middle of the oven and preheat to 425° F.

For the filling: Place 1 Tbsp vegetable oil into 10-inch skillet and set over high heat until it shimmers. Add the chicken, season with 1 tsp of salt, and cook, stirring occasionally, until the chicken is no longer pink on the outside but not dry, 4 to 6 minutes. Remove the chicken from the skillet and set aside ina medium bowl.

Decrease the heat to medium and heat the remaining Tbsp vegetable oil in the skillet until it shimmers. Add the onions, carrots, celery, garlic, remaining 1 1/2 tsp of salt, black pepper, dried thyme, and dried tarragon, and stir to combine. Cook, stirring occasionally, until the onions have softened, about 5 minutes. Add the butter and melt. Stir inthe flour and cook for 1 to 2 minutes. Gradually stir in the chicken broth and milk. Bring toa simmer,stirring continually. Continue stirring until the sauce thickens, about 3 minutes. Remove the skillet from the heat and stir in the peas, thyme, tarragon, and reserved chicken. Cover and set aside.

For the crust: on a lightly floured surface, unfold the pastry dough and cut into 4 even squares (cut once horizontally and once vertically). Gently roll out each square, increasing the size by about 20 percent.

For the egg wash: Combine the beaten egg and water in a small dish and set aside.

To build the pie: Spray 4 (16-oz) ramekins generously with cooking spray. Create 4 pie crusts using the puff pastry squares and shaping them in 4 seperate (16-oz) ramekins. Divide the warm filling into the 4 ramekins. Use remaining puff pastry squares to cover the ramekins, pinch the edges and use a little water on each to bind. Press the edges with a fork to create a decorative border. Brush each crust with the egg wash. Cut a small slit in the top to allow the steam to escape during baking.

Bake for 15 minutes. Decrease the heat to 400° F and bake until the crust is puffed and golden brown and the filling is bubbly, another 15 to 20 minutes. Cool on a rack for 10 minutes before serving.