Step 1

Butter a 13x9" baking dish with unsalted butter. Season one 1½–2-lb. skinless, boneless turkey breast generously with kosher salt and freshly ground black pepper. Heat 2 Tbsp. extra-virgin olive oil in a large high-sided skillet over medium. Cook turkey until browned, 6–8 minutes per side. Transfer to a cutting board; let rest 10 minutes.

Step 2

Meanwhile, add 2 Tbsp. water to pan, scraping up any browned bits. Increase heat to medium-high, add 2 Tbsp. unsalted butter and remaining 2 Tbsp. extra-virgin olive oil, and cook until butter is melted. Add 1 large onion, thinly sliced, and 1 large russet potato, scrubbed, cut into ½" pieces, season with kosher salt, and cook, stirring occasionally, until onion is softened and golden brown around edges, about 5 minutes. Add 20 oz. frozen mixed vegetables and 2 garlic cloves, finely chopped; season with kosher salt and pepper. Cook, stirring occasionally, until heated through, 5–7 minutes.

Step 3

Meanwhile, chop turkey into ½" pieces. Whisk together ¼ cup all-purpose flour and 2 cups water in a large measuring glass or small bowl.

Step 4

Reduce heat to medium and pour flour mixture into pan; stir vigorously until no lumps remain. If mixture seizes, add more water ¼-cupful at a time, whisking until sauce is smooth. Bring to a simmer and cook, stirring, until thickened, about 3 minutes. Remove pan from heat, add 4 oz. American cheese (4–6 slices), finely chopped, and stir until melted. Transfer turkey and any accumulated juices to pan, add ⅓ cup chopped parsley, and stir to combine. Scrape into prepared baking dish.

Step 5

Preheat oven to 400°. Unfold 1 sheet frozen puff pastry (one 14-oz. package or half of a 17.3-oz. package), thawed, and gently roll out on a lightly floured surface to smooth out creases and make slightly larger than baking dish if needed. Drape over dish and press gently so it touches filling. Brush with 1 large egg, beaten to blend, and sprinkle with flaky sea salt. Using a small knife, make at least four 1–2" slits in pastry for venting as desired.

Step 6

Bake pot pie until pastry is golden brown, 40–45 minutes. Let cool 20–25 minutes before serving.