Preheat the oven to 375 degrees.
In a nonstick pan on medium heat cook the ground beef, breaking it up into finer pieces using a spatula for 3 - 4 minutes. Add in the garlic and onion; cook for 3 - 4 minutes more.
Drain half of the grease out of the pan. Mix flour in well and cook for 1 minute. Stir in beef broth, season with salt and pepper. Once the broth starts to thicken add in the peas. Remove from heat and set aside.
Grease an appropriately sized baking dish or casserole dish with the butter. Place a layer of sliced potatoes on the bottom and half of the milk. Spread half of the beef mixture on the potatoes, top with 1/3 of the cheese. Do another layer of potatoes, remaining beef mixture, and 1/3 cheese; then a top layer of potatoes and the remaining milk. Season with salt and pepper and cover tight with foil.
Bake for 30 minutes or until the potatoes are tender (will depend upon how thick the potatoes are sliced). Then remove the foil, top with the remaining cheese and bake uncovered for another 5 minutes.
Remove the dish from the oven and let it sit for 5 - 10 minutes before cutting and serving with crusty bread on the side.