Preparation

• Heat the oven to 400°F. Wrap the beets in aluminum foil and roast until they’re just barely cooked through, about 50 minutes (you should be able to poke a fork in about 1/2 inch with some resistance). Reduce the oven temperature to 375°F.
• When the beets are at room temperature, trim off their ends, peel, and slice into 1/2-inch-thick rounds (you should yield about 6 slices). Lay the slices on a large surface and season both sides evenly with a total of about a scant 1/2 tsp. salt and a pinch of pepper.
• Meanwhile, melt 3 Tbs. butter in a 12-inch, heavy, nonstick frying pan over medium-high heat. When melted, add the shallots and sauté until translucent and aromatic, about 3 minutes. Add the mushrooms, thyme, 1 tsp. salt, and a large pinch of pepper, and sauté until the mushrooms are tender and all of the liquid has evaporated, about 13 minutes. Add 1/4 cup wine and simmer until all of the liquid has evaporated, about another 7 minutes. Remove and discard the thyme sprigs, and set the mushrooms aside off the heat to cool to room temperature (you can freeze them for about 15 minutes to expedite this process). Meanwhile, mix the soft goat cheese with the 1/4 cup chives, the zest, 1/4 tsp. salt, and a large pinch of pepper.
• Sprinkle about half of the flour on a large clean work surface. Unfold the puff pastry onto the surface and evenly and lightly flour both sides. Carefully stretch it a bit so it measures 12-1/2 x 12-1/2 inches square. Working about 5 inches down the center of the dough rectangle (leave about 3 inches each at the top and bottom and roughly 4 inches each at both sides), evenly arrange half of the mushroom mixture, then the beet slices, then the cheese mixture, and then the remaining mushroom mixture. Fold both ends of the dough over the edges of the vegetables to cover. Then fold both sides of the dough toward the center to enclose the filling completely; shape into an even rectangle. Carefully transfer the filled pastry to a parchment paper–lined baking sheet with sides. Brush the top and sides of the pastry with the egg wash. Refrigerate for 45 minutes to 2 hours.
• Bake in the center of the 375°F oven until the pastry is golden brown, puffed, and completely firm when poked, about 50 minutes. Let sit until you’ve made the sauce.
• Pour the remaining 2 cups wine into a small, heavy saucepan; place the dried mushroom sack into the wine and cook over high heat until the liquid is reduced by about half, about 16 minutes. Remove and discard the mushroom sack. Stir in the arrowroot-water mixture, the broth, sugar, the remaining 1/2 tsp. salt, and the remaining pepper; lower the heat to medium and simmer (do not boil) until thickened, about 6 minutes. Whisk in the remaining 5 Tbs. butter. Taste and adjust the seasonings if necessary (you should yield about 2 cups sauce).
• Slice the Wellington, carefully transfer to a serving platter, and garnish with the remaining chives. Serve the sauce in a gravy boat alongside the pastry or spoon some atop each serving.

Make Ahead Tips

Plan ahead by defrosting your puff pastry; it takes 2 to 3 hours in the fridge or 1 hour at room temperature. Prepare the mushroom and goat cheese fillings and roasted beets a day ahead of time.