Prepare the rabbit meat by trimming and cutting it into smaller pieces. Season generously with salt and pepper.
In a large skillet or Dutch oven, 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, garlic, carrots, and parsnips. In the same skillet, sauté the vegetables in the remaining butter until softened, about 5-7 minutes.
Sprinkle the flour over the vegetables and stir well to combine. Gradually add the white wine, stirring constantly to prevent lumps. Pour in the chicken stock, add the thyme sprigs and bay leaf. Bring to a simmer and cook for 15 minutes, allowing the sauce to thicken slightly.
Return the rabbit meat to the skillet with the vegetable sauce. Simmer for another 10 minutes, allowing the flavors to meld. Remove the thyme sprigs and bay leaf before proceeding.
Transfer the rabbit and vegetable mixture to a pie dish. Cover with the puff pastry, trimming off any excess. Crimp the edges to seal and prick the top 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.
Serve the rabbit pie hot, accompanied by a fresh green salad or steamed vegetables.
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