1.Preheat oven to 400°. Make mashed potatoes: In a large pot, cover potatoes with water and add a generous pinch of salt. Bring to a boil and cook until totally soft, 16 to 18 minutes. Drain and return to pot.
2.Use a potato masher to mash potatoes until smooth. Add melted butter, milk, and sour cream. Mash together until fully incorporated, then season with salt and pepper. Set aside.
3.Make beef mixture: In a large, ovenproof skillet over medium heat, heat oil. Add onion, carrots, garlic, and thyme and cook until fragrant and softened, 5 minutes. Add ground beef and cook until no longer pink, 5 minutes more. Drain fat.
4.Stir in frozen peas and corn and cook until warmed through, 3 minutes more. Season with salt and pepper.
5.Sprinkle meat with flour and stir to evenly distribute. Cook 1 minute more and add chicken broth. Bring to a simmer and let mixture thicken slightly, 5 minutes.
6.Top beef mixture with an even layer of mashed potatoes and bake until there is very little liquid visible and mashed potatoes are golden, about 20 minutes. Broil if desired.
7.Garnish with parsley before serving.