Prepare the rabbit meat by trimming it and cutting it into bite-sized pieces. Season generously with salt and pepper.
In a large skillet, melt the butter over medium-high heat. Sear the rabbit meat until golden brown on all sides. Remove from the skillet and set aside.
Finely chop the onion, leeks, and carrots. Add them to the skillet and sauté until softened.
Sprinkle the flour over the vegetables and stir to combine. Gradually pour in the chicken stock, stirring constantly to prevent lumps. Add the thyme sprigs and bay leaf.
Return the rabbit meat to the skillet. Reduce the heat to low, cover, and simmer for 20 minutes, or until the flavors have melded together.
Transfer the mixture to a pie dish. Top with the puff pastry, trimming any excess. Score the pastry with a fork to allow steam to escape.
Whisk the egg and brush it over the top of the pastry. Bake in a preheated oven at 400°F (200°C) for 25-30 minutes, or until the pastry is golden brown and puffed up.
Let the pie rest for 10 minutes before slicing and serving hot.
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