In a small metal bowl, whisk together egg yolk, garlic, small pinch of salt, and cold water. Whisk until well blended, then drizzle in a little bit of olive oil, and continue to whisk. Continue to drizzle olive oil in until fully integrated and sauce is thick and emulsified. Whisk in miso paste and lemon juice. Season with salt to taste.
To make the potatoes:
Place shredded potatoes in a bowl of cold water and stir until water is cloudy and starchy. Next, drain the potatoes. Working in batches, transfer potatoes to a clean kitchen towel and squeeze and twist the towel over the sink to get as much liquid out as possible (this will help you get your hash browns crispy). Return to dry bowl, season with salt and pepper.
Melt 4 tablespoons butter in a cast iron pan over low heat, being careful not to brown.
Once butter is melted, turn it up to medium-high. Add potatoes and spread evenly. Cook without stirring for at least 5 minutes, until underside has started to brown. Gently break up potatoes and flip them over, and cook for an additional 8-10 minutes, turning once or twice until most hash browns are nicely browned on both sides. If potatoes are too dry, add in addition 1-2 tablespoons melted butter.
Drizzle miso aioli over hash browns and top with scallions.