Cut a sheet of parchment paper to match the outline of the Dutch oven lid and place it on a large plate or upturned baking sheet. Roll out the puff pastry sheet into a 15x11-inch rectangle on a lightly floured surface. Using a pizza cutter or sharp knife, cut the pastry widthwise into ten 1 ½-inch-wide strips.
Space 5 pastry strips parallel and evenly across the parchment circle. Fold back the 1st, 3rd, and 5th strips. Lay one additional pastry strip perpendicular to the 2nd and 4th strips, keeping it snug to the edges of the pastry, then unfold the original strips. Repeat with the remaining pastry strips, alternating between folding back different sections to create a lattice pattern. Trim any excess pastry along the outline of the parchment circle. Cover loosely with plastic wrap and refrigerate while preparing the filling.
Adjust the oven rack to the lower-middle position and preheat the oven to 400°F. Melt butter in a Dutch oven over medium heat. Add leeks, carrots, and salt, cooking until softened (about 5-7 minutes). Stir in garlic and tomato paste and cook until fragrant, about 1 minute. Stir in flour and cook for 1 more minute.
Gradually stir in chicken broth, scraping up any browned bits and smoothing out lumps. Stir in heavy cream, soy sauce, and bay leaves. Bring to a simmer and cook until the mixture thickens, about 3 minutes. Stir in the chicken pieces and return to a simmer.
Turn off the heat, cover the Dutch oven with the inverted lid, and carefully place the parchment with the prepared pastry lattice on top. Brush the pastry with egg wash and sprinkle with salt. Transfer the pot to the oven and bake until the pastry is puffed and golden brown, about 25-30 minutes, rotating the pot halfway through baking.
Remove the pot from the oven and transfer the parchment with the baked pastry to a wire rack. Discard the bay leaves. Stir asparagus into the filling and cook over medium heat until crisp-tender, about 3-5 minutes. Turn off the heat, stir in peas and let sit until heated through, about 5 minutes. Adjust the consistency with extra hot broth as needed. Stir in tarragon, lemon zest, and juice. Season with salt and pepper to taste. Place the baked pastry back on top of the filling and serve warm.