Preheat oven to 425°F / 218°C.
Line a sheet pan with parchment paper (this will help to prevent sticking).
Spread hash browns out on prepared sheet pan. Drizzle melted butter over top and stir gently. Season with some salt and pepper (see note).
Transfer pan to the oven and bake until hash browns are golden brown, 35 to 40 minutes. Halfway through cooking, remove the pan from the oven and gently turn the hash browns with a spatula so that the crispy edges are rotated to the center and the center potatoes are moved to the outside of the pan. (The edges will get golden brown and crisp, so you want to give most of the potatoes a “turn” to cook on the outside of the pan.)
Meanwhile make salsa by combining avocados, tomatoes, cilantro, shallots, and lime juice. Season with some salt and pepper. (If you’d like it to be spicy, you can add some hot sauce or a pinch of red pepper flakes.)
Once hash browns are golden brown remove the pan from the oven. (Note: If they aren’t golden brown in most places at the end of 35 to 40 minutes, you can continue baking them a bit longer. Be sure that they are mostly crisp and golden brown before adding the eggs.
Use the back of a spoon to create 6 wells in the hash browns.
Crack eggs into the wells. Return pan to the oven and continue cooking until the eggs are cooked to your liking (about 5 minutes if you’d like the yolks soft and 8 minutes if you’d like them to be cooked all the way through).
Serve immediately with salsa on top.