Brown the meat: Heat oil in a heavy pot over medium-high heat. Brown the venison in batches until well seared. Remove and set aside.
Build the base: In the same pot, sauté onions until soft and golden. Add garlic and carrots and cook for 2 to 3 minutes.
Make the gravy: Stir in tomato paste and flour and cook for about 1 minute. Gradually pour in the red wine and stock, stirring to prevent lumps.
Slow cook: Return the venison to the pot. Add bay leaves, thyme, coriander, chutney, salt, and pepper. Cover and simmer gently for 1½ to 2 hours until the meat is tender and the sauce has thickened. Remove the bay leaves and allow the filling to cool completely.
Assemble: Preheat the oven to 180°C. Spoon the cooled filling into a pie dish. Cover with puff pastry, trim the edges, and press to seal. Cut a few small slits in the top for steam to escape. Brush with beaten egg.
Bake: Bake for 25 to 35 minutes until the pastry is golden and crisp.