Get ready to create some deliciousness by preheating the oven to 400°. In a medium Dutch oven or other heavy pot, melt 3 Tbsp. unsalted butter over medium-high heat. Sprinkle 3 Tbsp. all-purpose flour 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 and coarsely chopped, 2 celery stalks, thinly sliced, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt. Cook, stirring occasionally, until the carrots are slightly softened, 5–7 minutes.
Step 2
Next, add 3 garlic cloves, thinly sliced, and 2 tsp. thyme leaves to the pot; season generously with freshly ground pepper. Cook and stir until the garlic is fragrant, about 1 minute. Pour in 1 cup dry white wine and cook, stirring often, until evaporated and the mixture is pasty, about 1 minute. Add 1 lb. small potatoes, scrubbed and 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.
Step 3
While that simmers, roll out 1 sheet frozen puff pastry (one 14-oz. package or half of a 17.3-oz. package), thawed, on a surface (no need to flour) to a 16×12″ 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.
Step 4
Using a potato masher or the back of a wooden spoon, mash about one third of the potatoes in the pot so they release their starch. Add 2 cups shredded cooked chicken meat, 1 cup heavy cream, and ½ cup frozen peas to the soup. Cook, stirring occasionally, until the chicken and peas are heated through and the soup is slightly thickened, about 5 minutes. Taste the soup and season with more kosher salt if needed.
Step 5
Ladle the soup into bowls and top with chopped parsley and more pepper. Tear the puff pastry into large pieces; serve alongside.
Do Ahead: Soup can be made 2 days ahead. Let cool; cover and chill. Puff pastry can be baked 1 day ahead; store airtight at room temperature.