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Add potatoes to pot; cover with water and season generously with sea salt. Bring to boil; then cover until mostly cooked, about 7 minutes. Drain well and allow to cool.
Meanwhile, drizzle a little oil onto large skillet and cook chorizo 7 minutes over medium heat until brown and rendered of all fat. Use a slotted spoon to move chorizo to a plate, leaving as much of fat as possible in skillet. There should be about 2 tablespoons; if not, add a bit of oil; set skillet aside.
Preheat oven to 425 degrees.
Cut potatoes into ½‑inch thick rounds. Heat fat left in skillet. Add potatoes in as close to a single layer as possible. Stir in paprika and chili powder and cook 8 minutes, tossing occasionally, until potatoes begin to brown. Season with sea salt and pepper. Add mushrooms and onion and cook 3 minutes, until onions brown slightly and potatoes get crusty. Return chorizo to pan.
Remove skillet from heat and make 4–6 small, evenly-spaced wells in hash, exposing bottom of skillet. Pour a small bit of olive oil in each well and break an egg into it; sprinkle with sea salt and black pepper. Transfer skillet to oven and bake until eggs are cooked to your taste, about 5 minutes for firm whites and soft yolks. Serve immediately with desired toppings, like fresh cilantro, sour cream and/or salsa.