Roll out each puff pastry sheet to an even thickness. Cut two 8-inch rounds from each sheet using a sharp knife.
Line bottom and sides of two 8-inch pie tins with pastry rounds. Reserve 2 pastry rounds for tops of pies. Place pie tins and tops on a baking sheet; transfer to the refrigerator.
Cook and stir sausage in a skillet over medium heat until browned, about 5 minutes. Drain excess grease. Add butter, carrots, leek, and garlic; cook and stir until leek softens, about 5 minutes.
Stir mushrooms into the skillet; cook until tender, about 8 minutes. Stir in flour for 1 minute. Pour in vegetable broth slowly; add Dijon mustard, cayenne pepper, nutmeg, salt, and pepper. Cook and stir until sauce thickens, about 5 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Divide sausage mixture evenly between the pie tins. Cover with pastry tops.
Whisk egg and water together in a small bowl to make egg wash. Brush egg wash over pastry tops.
Bake in the preheated oven until golden brown, about 35 minutes. Cool briefly before slicing, about 10 minutes.