Step 1
Place a rack in middle of oven; preheat to 400°. Toss 1 lb. skinless, boneless chicken breasts (about 2 medium) with 1 Tbsp. green curry paste, 1 Tbsp. vegetable oil, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt on a large rimmed baking sheet until evenly coated. Bake until chicken is opaque and just cooked through (an instant-read thermometer inserted in the thickest part of a breast should register 150°), 15–20 minutes. Set chicken aside; leave oven on.
Step 2
Heat 4 Tbsp. unsalted butter in a large Dutch oven or other heavy pot over medium. Add 1 large onion, cut into ½” pieces, 1 large or 3 medium carrots, peeled, halved lengthwise, thinly sliced, 3 baby Yukon Gold potatoes, scrubbed, quartered, thinly sliced, and 4 garlic cloves, finely chopped. Cover and cook, stirring occasionally, until onion is translucent and carrots are beginning to soften, 8–10 minutes. Sprinkle ¹⁄₃ cup all-purpose flour over and cook, stirring constantly, until flour is pale golden, about 5 minutes.
Step 3
Add remaining 3 Tbsp. green curry paste and 1¾ cups water to pot; stir to combine. Bring mixture to a boil, reduce heat, and simmer, stirring constantly, until sauce is very thick and potatoes are nearly tender, 6–8 minutes. Stir in ½ cup heavy cream and bring to a simmer; remove from heat.
Step 4
Transfer reserved chicken to a cutting board and cut into ½” pieces. Add to filling in pot along with 1 cup frozen green peas, 2 Tbsp. fresh lime juice, 1 Tbsp. fish sauce, and 1 tsp. sugar; stir to combine. Taste filling and season with salt if needed. Scrape into an 8×8″ baking dish or a 9″-diameter deep glass pie dish; smooth surface.
Step 5
If using a 14-oz. package of frozen puff pastry, cut dough in half and gently roll out 1 half on a lightly floured surface just to smooth any creases and make slightly larger than the baking dish; reserve remaining half for another use.
Step 6
If using a 17.3-oz. package of frozen puff pastry, gently roll out 1 sheet on a lightly floured surface to smooth out any creases and make slightly larger than the baking dish.
Step 7
Lightly brush puff pastry with cream, then punch out a 1″-diameter round from the center with a small cutter or bottle cap (or cut out with a paring knife). Drape pastry over baking dish, letting it hang a little over the edges (do not crimp). Trim, leaving about a ¼” overhang if needed.
Step 8
Place baking dish on a rimmed baking sheet (to catch drips) and bake pot pie until puff pastry is golden brown and filling is bubbling around the edges, 25–30 minutes. Let cool 10 minutes before serving.