Cook the bacon in a large nonstick skillet over medium heat until crisp, 7 to 9 minutes, then transfer it to a paper towel and leave 2 tbsp fat in the pan.
Spread the hash browns in the skillet, season with 1/4 tsp salt and the pepper and press them into an even layer.
Cook without disturbing until the bottom is deeply browned, 6 to 8 minutes, then turn in sections and brown the other side for 4 minutes.
Whisk the eggs, milk and remaining 1/4 tsp salt until even in color.
Melt the butter in a second skillet over medium-low heat, add the eggs and fold slowly until soft curds form and no liquid egg remains, 3 to 4 minutes.
Divide the potatoes among four bowls and top with the eggs, cheddar, bacon, tomato and scallions.