Prepare campfire and let the coals get hot before beginning to cook. Set the Dutch oven over the hot coals.
Add the oil and let it heat for 1 minute. Stir in the beef or lamb and season with salt and pepper. Cook, breaking it up with a wooden spoon until browned, 7 to 8 minutes.
Add the onion, carrot, and celery and cook until softened, stirring frequently, 9 to 10 minutes.
Stir in the tomato paste and cook for 2 minutes. Sprinkle the flour over the surface of the meat and mix to combine. Pour in the broth.
Bring the meat mixture to a simmer and cook until thickened, about 2 minutes. Taste and season with salt and pepper, if needed.
Remove the Dutch oven from the heat and carefully spread the potatoes over the meat mixture, covering the entire surface.
Place the lid on the Dutch oven and set it over the hot coals. Place 12 hot coals or embers on the lid of the Dutch oven. Cook until the potatoes are heated through and the meat mixture is bubbling up around the edges of the potatoes, about 30 minutes.
Let the shepherd's pie stand for 10 minutes before serving.