Make the pico de gallo by combining the first 7 ingredients in a medium-sized bowl. Stir well, cover, and refrigerate until ready to use. This can be made several days in advance.
Remove the casings from the chorizo and add to a skillet over medium-high heat. Cook for 4-5 minutes, stirring occasionally. Turn off the heat and cover to keep warm.
Peel the potato. Place a clean tea towel on the counter and shred the potato onto the towel, using the largest holes of the grater. Bundle up the ends of the towel and wring out as much moisture from the potatoes as possible. (Once peeled and grated, don't let them sit too long as they will begin to turn brown).
In a medium non-stick skillet, melt 1 tablespoon of butter over medium-high heat. Add half of the shredded potatoes and spread out to a 6-inch circle. Use a wide, flat spatula to press down firmly on the potatoes. Season with salt and pepper (go light on the salt, as the chorizo will add a bit of salt to the overall dish). Reduce the heat to medium and continue cooking, pressing down with the spatula occasionally. Cook for 4 -5 minutes or until golden-brown and crisp. Carefully transfer to a plate, cooked side down and add another tablespoon of butter to the skillet. Once the butter has melted, turn the hash brown back into the pan and cook the other side, pressing it down with a spatula. (If a piece breaks off, just press it back into place). Cook for 3 - 4 minutes until golden-brown and crisp. Transfer the hash brown to a plate and cook the other half of the shredded potatoes in the same manner. (If you have an extra skillet on-hand, you can cook both hash browns at the same time to expedite the process).
In a non-stick skillet over medium heat, add the last tablespoon of butter. Once the butter has melted, crack the eggs into the pan and immediately cover with a lid. Cook until the egg white is cooked through and the egg yolk is barely set, approximately 5 minutes.
Top each hash brown with chorizo, an egg, pico de gallo, sliced avocado, and crumbled queso fresco. Serve immediately and enjoy!