Preheat the oven to 400°F (200°C).
Heat 2 tablespoons of olive oil in a large 12-inch ovenproof skillet over medium-high heat.
Add half of the mushrooms and season with salt and pepper. Cook without stirring for about 3 minutes, until browned. Stir and continue cooking for another 3 minutes, or until browned on multiple sides. Transfer to a bowl.
Add the remaining 2 tablespoons of olive oil and cook the remaining mushrooms in the same way. Transfer them to the bowl with the first batch.
Reduce the heat to medium and melt the butter in the skillet.
Add the onion and carrots. Cook for about 5 minutes, stirring occasionally, until they begin to soften.
Add the garlic and cook for about 30 seconds, until fragrant.
Stir in the tomato paste and cook for 1 minute.
Sprinkle in the flour and cook for another minute, stirring constantly.
Add the kale, vegetable stock, thyme, and soy sauce. Season with salt and pepper. Cook for about 3 minutes, until the kale begins to soften.
Stir in the heavy cream and cook until the mixture becomes creamy and slightly thickened.
Return the browned mushrooms to the skillet and stir until everything is evenly combined. Taste and adjust with additional soy sauce, salt, and black pepper as needed.
Roll the puff pastry out on a lightly floured surface until it is large enough to cover the skillet. Trim it into a rough circle if necessary.
Carefully place the pastry over the mushroom filling. Crimp the edges with a fork if desired.
Gently score a decorative pattern across the pastry using a paring knife, being careful not to cut completely through the dough.
Whisk the egg with 1 tablespoon of heavy cream. Lightly brush the mixture over the puff pastry.
Place the skillet on a baking sheet to catch any bubbling filling. Bake for about 30 minutes, or until the pastry is puffed and deeply golden.
Remove from the oven and let the Mushroom Pot Pie rest for 10 minutes before serving.