Preheat oven to 375°F. Boil potatoes and carrots in well-salted water until they are just tender; avoid overcooking as they will be cooked again in the oven. When done, drain water and set side.

In a 12-inch sauté pan over medium heat, heat the oil. Add leeks, fennel, mushrooms, and salt. Cook while stirring occasionally, for about 10 to 15 minutes, until vegetables have softened and mushrooms have browned. In a small bowl, stir together the cornstarch and about 1/4 cup of the broth to make a slurry. Add lemon juice, mustard, and pepper. Pour the slurry and remaining vegetable broth into the pan and stir. Continue cooking until the sauce reaches a low simmer and thickens, about 3 to 5 minutes: Sauce should coat the back of a spoon when done. Fold in cooked potatoes and carrots, peas, tarragon, and thyme. Taste and season with more salt and pepper, if desired. Remove from the heat and set aside.

Remove puff pastry from the package and trim to fit a 9-inch square casserole dish, leaving about 1/4-inch extra on all sides for crimping. Transfer vegetable mixture from the pan to the casserole. Place the puff pastry on top of the filling and gently crimp around the edges. Use a cookie cutter to cut shapes from any extra pastry and use them to decorate the top of the casserole. With a sharp knife, cut several small slits in the pasty to release steam while cooking. Place the casserole on top of a baking sheet lined with parchment. Bake until the pastry is puffed and golden brown, about 30 minutes. Let stand at room temperature for 10 minutes before serving.

Making these in advance? Unbaked potpies can be refrigerated for up to a day before baking. Increase baking time to 40 minutes.