Prepare the vegetables. Peel and chop the carrots, parsnip, and rutabaga into small bite-sized pieces. Slice the leeks and quarter the mushrooms. Dice the onions and mince the garlic. Set the vegetables aside.
Bring a pot of water to a boil and generously season with kosher salt. When the water comes to a boil add the parsnips and cook until fork tender. Use a slotted spoon to remove the parsnips. Repeat with both the rutabaga and the carrots.
Meanwhile add the butter to a deep skillet and heat over medium heat. Add the leeks and onions to the butter and sauté until softened. Add the mushrooms and continue cooking until the mushrooms are cooked through. Add the garlic and fresh thyme and sauté for 30 seconds or until fragrant.
Add the flour to the skillet and stir until completely blended with the butter. Slowly add the vegetable broth to the skillet, stirring constantly to prevent lumps. Bring the sauce to a simmer, stirring occasionally to prevent sticking.
Stir the crème fraiche into the sauce and add the shredded cheese and grainy mustard. Cook, stirring occasionally until the cheese is melted. Stir in the chopped fresh parsley. Add the cooked root vegetables and set aside to cool.
Preheat the oven to 400°F / 204°C. On a lightly floured counter unroll the puff pastry. Lightly roll the puff pastry to make it slightly bigger. If using individual ramekins, cut out pieces of pastry that fit the top of the ramekins leaving a half inch more on each side. If using a baking dish, leave the pastry whole.
Transfer the cooled filling to the individual ramekins or a large baking dish. Brush the edge of the dish with the beaten egg. Lay the puff pastry on top of the dish and press around the edge of the dish to seal.
Brush the top of the pastry with the beaten egg. Cut several slits on top of the pies. Sprinkle the top with chopped fresh thyme and coarse sea salt for garnish if desired.
Place the ramekins or baking dish on a baking sheet to catch any overflow. Bake the pie for 30 to 40 minutes or until the top is golden brown and the filling is bubbling. Let cool slightly before serving and enjoy!