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.
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.
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.
Stir in frozen peas and corn and cook until warmed through, 3 minutes more. Season with salt and pepper.
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.
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.