Total time 1 hour 30 minutes
1. Cut sheet of parchment paper to match outline of Dutch oven lid and place on large plate or upturned rimmed baking sheet. Roll puff pastry sheet into 15 by 11‑inch rectangle on lightly floured counter. Using pizza cutter or sharp knife, cut pastry widthwise into ten 11/2‑inch-wide strips.
2. Space 5 pastry strips parallel and evenly across -parchment circle. Fold back first, third, and fifth strips almost completely. Lay additional pastry strip perpendicular to second and fourth
strips, keeping it snug to folded edges of pastry, then unfold strips. Repeat laying remaining 4
pastry strips evenly across parchment circle, alternating between folding back second and
fourth strips and first, third, and fifth strips to create lattice pattern. Using pizza cutter, trim edges of pastry following outline of parchment circle. Cover loosely with plastic wrap and refrigerate while preparing filling.
3. Adjust oven rack to lower-middle position and heat oven to 400 degrees. Melt butter in Dutch oven over medium heat. Add leeks, carrots, and 1 teaspoon salt and cook until vegetables are softened, about 5 minutes. Stir in flour, garlic, and tomato paste and cook for 1 minute.
4. Slowly stir in broth, scraping up any browned bits and smoothing out any lumps. Stir in cream, soy sauce, and bay leaves. Bring to simmer and cook until mixture is thickened, about 3 minutes. Stir in chicken and return to simmer.
5. Off heat, cover pot with inverted lid and carefully place parchment with pastry on lid. Brush pastry with egg and sprinkle with salt. Transfer pot to oven and bake until pastry is puffed and golden brown, 25 to 30 minutes, rotating pot halfway through baking.
6. Remove pot from oven. Transfer parchment with pastry to wire rack; discard parchment. Remove lid and discard bay leaves. Stir asparagus into filling and cook over medium heat until crisp-tender, 3 to 5 minutes. Off heat, stir in peas and let sit until heated through, about 5 minutes. Adjust filling consistency with extra hot broth as needed. Stir in tarragon and lemon zest and juice. Season with salt and pepper to taste. Set pastry on top of filling and serve.