Preheat oven to 400°.

Melt 3 Tbsp. unsalted butter in a medium Dutch oven or other heavy pot over medium-high heat.

Sprinkle 3 Tbsp. all-purpose flour over and cook, whisking constantly, until combined and mixture is light golden brown, about 3 minutes.

Add 3 large shallots, coarsely chopped, 2 medium carrots, peeled, coarsely chopped, 2 celery stalks, thinly sliced, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt.

Cook, stirring occasionally, until carrots are slightly softened, 5–7 minutes.

Add 3 garlic cloves, thinly sliced, and 2 tsp. thyme leaves to pot; season generously with freshly ground pepper.

Cook, stirring, until garlic is fragrant, about 1 minute.

Pour in 1 cup dry white wine and cook, stirring often, until evaporated and mixture is pasty, about 1 minute.

Add 1 lb. small potatoes, scrubbed, cut into ½" pieces, 7 cups low-sodium chicken broth, 2 Tbsp. Dijon mustard, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt and bring to a boil.

Reduce heat and simmer, stirring occasionally, until potatoes are tender, 20–25 minutes.

Meanwhile, roll out 1 sheet frozen puff pastry on a surface to a 16x12" rectangle; transfer to a parchment-lined baking sheet.

Brush with 1 large egg, beaten to blend, and season generously with flaky sea salt and pepper.

Prick pastry all over with a fork and bake until deep golden brown and crisp, 15–20 minutes.

Let cool slightly.

Using a potato masher or the back of a wooden spoon, mash about one third of potatoes in pot so they release their starch.

Add 2 cups shredded cooked chicken meat, 1 cup heavy cream, and ½ cup frozen peas to soup.

Cook, stirring occasionally, until chicken and peas are heated through and soup is slightly thickened, about 5 minutes.

Taste soup and season with more kosher salt if needed.

Ladle soup into bowls and top with chopped parsley and more pepper.

Tear puff pastry into large pieces; serve alongside.