Preheat oven to 400°F.
Melt the butter in a large pot over medium. Add the leeks and cook, stirring, 5 minutes. Add the carrots and mushrooms and cook, 5 minutes. Add the wine and cook, 2 minutes. Sprinkle with the flour; cook, stirring constantly, 3 minutes. Gradually whisk in 3 cups of the milk and simmer, whisking constantly, until thickened, 2 to 3 minutes. Stir in the peas, tarragon, salt, and pepper. Spoon the mixture into an 8-inch square baking dish.
Place the puff-pastry sheet over the filling, pressing the edges to seal. Brush with the remaining milk; cut slits into the top to release steam.
Bake until the pastry is golden brown and the filling is bubbly, 30 to 35 minutes. Let stand for 15 minutes before serving.