Preheat oven to 400 degrees F.

In a large saucepan or Dutch oven, melt the butter over medium-high heat. Add the chopped carrots, onion, and fennel, and cook about 2 minutes.

Stir in the potatoes, mushrooms, thyme, salt, and pepper. Cook for 5 minutes, stirring occasionally, until the mushrooms shrink and release their liquid. Stir in garlic and cook 1 minute more.

Sprinkle the flour over the vegetables and stir well. Cook for 1 minute, then slowly pour in the broth, followed by 1 cup of the milk, stirring constantly. Continue to stir until the liquid mixture is smooth. Bring it to a simmer and cook until it thickens a bit, about 5 minutes.

Remove the mixture from the heat and stir in the peas, parsley, and a generous squeeze of lemon juice. Be sure to taste at this point, and add more salt, pepper, or thyme as desired.

In a small bowl, whisk together the egg with 1 tablespoon whole milk to form an egg wash.

Scoop the pie filling out into an 8x8-inch baking dish. Carefully lay the puff pastry over the filling, trimming any excess from the edges. Cut slits in the top to form vents, and brush lightly with the egg wash.

Bake for 25-30 minutes, until the crust is golden brown. You may see the mixture bubbling around the edges. Remove from the oven and let cool for 5-10 minutes before serving. Enjoy!