This breakfasty/brunchy dish originated on camping trips. We would often have leftover potato packets or mashed potatoes and some version of meat (bbq ribs, grilled steak, chicken or fish) that would get assembled into patties the next morning for breakfast. The combinations are endless (white or sweet potatoes, just about any kind of meat, leftover mushrooms/greens/onions and cheese) and they make a filling dish for any meal of the day. One of my favorite combinations for breakfast is pastrami and green onions, but you can really tweak it to suit your ingredients and mood.
Peel and cut potatoes into about 4 equal pieces each. Place into a pot, cover with cold water, bring to a very gentle simmer and cook until tender. Mash or put through ricer with butter and cream. Stir in milk to desired consistency (add more if needed). Season to taste with salt and pepper.
Mix potatoes, green onions, cheese, 1 egg and pastrami until well combined. Melt butter in nonstick skillet over medium heat. Divide potatoes into four patties* and place in pan. Make a well in the center of each with the back of a spoon and crack an egg into each. Salt and pepper and cover. Cook until whites are just set.
You can try flipping patties if you wish or place them briefly under the broiler to crisp the top, but they will also be fine cooked on the one side.
* If you are feeling fancy, you can pipe the potatoes into little nests, though you either have to very finely chop the pastrami or serve it on the side.