Preheat the oven to 400°F (200°C). Clean the pigeon breasts and season generously with salt and pepper.
In a saucepan, melt the butter over medium heat. Sauté the finely chopped shallots and garlic until translucent and fragrant.
Add the chopped mushrooms and cook until they soften and their moisture has evaporated.
Pour in the red wine and simmer until the liquid has reduced by half. Add the chicken stock and thyme sprigs, and continue to simmer until the sauce thickens slightly.
Roll out the puff pastry and cut it into two pieces, one slightly larger than the other. Place the larger piece into a pie dish or on a baking sheet.
Layer the pigeon breasts and the mushroom mixture onto the pastry base. Cover with the remaining pastry, pressing the edges firmly to seal.
Brush the top of the pie with beaten egg and bake for 25-30 minutes, or until golden brown and the pastry is cooked through.
Let the pie rest for 10 minutes after baking before slicing and serving.
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