Heat the oil in a large skillet over medium-high. Add the ground beef and andouille, break it up with a wooden spoon, and cook 6 to 7 minutes until browned.
Stir in the onion, bell pepper and celery and cook 5 minutes until soft. Add the garlic and stir 30 seconds.
Sprinkle in the flour and stir 1 minute. Stir in the tomato paste, Worcestershire and Cajun seasoning, then the broth. Simmer 8 to 10 minutes, stirring, until thick enough to hold a trail. Season with salt and pepper.
Spread the filling on a plate and cool at least 20 minutes.
Heat the oven to 400°F with a sheet pan inside. Cut four 6 1/2-inch rounds from the pie crusts, press them into four 5-inch pie tins and prick the bottoms with a fork.
Spoon the cooled filling into the crusts, leveling it just below the rim.
Cut four puff pastry rounds slightly larger than the tops. Brush the rims with egg, lay on the lids and crimp with a fork. Cut a small steam slit and brush the tops with egg.
Set the pies on the hot sheet pan and bake 25 to 30 minutes until puffed and deep golden. Rest 5 minutes, then lift out with a spatula.
Zigzag Louisiana hot sauce over the top and scatter on green onion. Serve hot.