Preheat oven to 400 degrees. Unfold 1 thawed sheet (half of a 17.3-ounce box) frozen puff pastry; roll out on a lightly floured surface until about 11 by 15 inches and 1/8 inch thick. Score with a sharp paring knife in several places. Transfer to a baking sheet; refrigerate at least 30 minutes (and up to 3 hours).
Meanwhile, heat butter in a large saucepan over medium-low. Add onion and carrots; cook, stirring occasionally, until softened, about 10 minutes. Add flour; cook, stirring, 2 minutes. Add wine, 2 cups water, and tarragon; season generously with salt and pepper. Stir briskly to prevent any lumps from forming.
Bring to a boil. Add beans and squash; reduce heat. Cover; simmer, stirring occasionally, until beans are crisp-tender, 8 to 10 minutes. Stir in 3/4 cup cream and chicken; remove from heat. Cover to keep warm.
Remove pastry from refrigerator; brush with remaining tablespoon cream. Bake until golden brown, 12 to 15 minutes. Transfer to a cutting board; cut into 8 pieces with a serrated knife.
Meanwhile, return chicken mixture to a simmer. Ladle into 8 dishes; top with pastry, and serve immediately.