Cook the ramen noodles according to package directions. Drain and cut into 1- to 2-inch pieces.
Roll out the pie crusts to a thickness of 1/4 inch. Using an 8-ounce ramekin as a guide, cut out 4 circles with an extra 1-inch border of crust all around.
Preheat the oven to 425 degrees F.
Bring the chicken stock to a boil in a medium saucepan. Blanch the carrots, peas and leeks separately in the broth until tender and set aside. Reserve the broth.
Melt the butter over a medium flame in a saucepan. Add the flour and cook until it begins to smell toasted, 1 to 2 minutes. Add the milk and cream, and whisk until incorporated and smooth. Add the reserved chicken stock and bring to a simmer; season with salt and pepper. Cook the mixture until thickened, and then add the chicken, peas, carrots, leeks, noodles and chives.
Divide this mixture among four 8-ounce ramekins, top with the crusts, fold the borders over the sides and cut vents in the tops. Brush with the beaten egg and season with salt.
Bake until the pastry is golden and bubbly, 15 to 20 minutes. Allow to cool slightly before serving.