Melt the butter in a medium pot over medium heat. Add the carrot, celery and onion and cook, stirring occasionally, until the vegetables begin to soften, 3 to 4 minutes. Stir in the thyme and cook until fragrant, about 1 minute. Sprinkle in the flour and cook, stirring, until the vegetables are coated, about 1 minute. Gradually pour in the chicken broth, stirring constantly until smooth. Stir in the cream, 1 teaspoon kosher salt and a few grinds of black pepper and bring to a boil. Lower the heat and simmer gently until thickened and the flour flavor is gone, about 10 minutes. Stir in the chicken and cook until just cooked through, about 3 minutes. Remove from the heat and stir in the peas. Transfer the filling to a medium bowl and let cool completely before assembling the hand pies.

Preheat the oven to 400 degrees F. Line a baking sheet with parchment and lightly coat with cooking spray.

Unfold the puff pastry sheets on a lightly floured surface and roll each sheet gently to smooth the seams. Cut each sheet into 4 equal rectangles, for 8 rectangles total.

Divide the cooled filling evenly onto one half of each pastry rectangle, leaving a 1/2-inch border around the edges. Brush the borders lightly with the egg wash, fold the short side of the pastry over the filling to form a smaller rectangle and press the edges with a fork to seal. Arrange the 8 hand pies on the prepared baking sheet, brush the tops with the remaining egg wash and sprinkle with flaky sea salt and smoked paprika. Place the hand pies in the freezer for 15 minutes to firm up.

Cut 2 to 3 small vents into the top of each hand pie. Bake until puffed and deep golden brown, 35 to 40 minutes. Let cool for 10 minutes, then serve.